Measurement, Perimeter, Area, and Volume

Make Unit Conversions in the Imperial System

There are two systems of measurement commonly used around the world. Most countries use the metric system. Canada uses the metric system, and the United States use the imperial system of measurement. However, people in Canada often use imperial measurements as well. We will look at the imperial system first.

The imperial system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart, and gallons. Both the imperial system and the metric system measure time in seconds, minutes, and hours.

The equivalencies of measurements are shown in the table below. The table also shows, in parentheses, the common abbreviations for each measurement.

Imperial System of Measurement

In many real-life applications, we need to convert between units of measurement, such as feet and yards, minutes and seconds, quarts and gallons, etc. We will use the identity property of multiplication to do these conversions. We’ll restate the identity property of multiplication here for easy reference.

To use the identity property of multiplication, we write 1 in a form that will help us convert the units. For example, suppose we want to change inches to feet. We know that 1 foot is equal to 12 inches, so we will write 1 as the fraction . When we multiply by this fraction we do not change the value, but just change the units.

But also equals 1. How do we decide whether to multiply by or ? We choose the fraction that will make the units we want to convert from divide out. Treat the unit words like factors and “divide out” common units like we do common factors. If we want to convert inches to feet, which multiplication will eliminate the inches?

The inches divide out and leave only feet. The second form does not have any units that will divide out and so will not help us.

When we use the identity property of multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.

EXAMPLE 2

A female orca in the Salish Sea weighs almost 3.2 tons. Convert her weight to pounds.

 

Solution

We will convert 3.2 tons into pounds. We will use the identity property of multiplication, writing 1 as the fraction .

Multiply the measurement to be converted, by 1.
Write 1 as a fraction relating tons and pounds.
Simplify.
Multiply.
	The female orca weighs almost 6,400 pounds.

Sometimes, to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.

EXAMPLE 3

Juliet is going with her family to their summer home. She will be away from her boyfriend for 9 weeks. Convert the time to minutes.

Solution
To convert weeks into minutes we will convert weeks into days, days into hours, and then hours into minutes. To do this we will multiply by conversion factors of 1.

	9 weeks
Write 1 as , , and .
Divide out the common units.
Multiply.
Multiply.

Juliet and her boyfriend will be apart for 90,720 minutes (although it may seem like an eternity!).

EXAMPLE 4

How many ounces are in 1 gallon?

Solution

We will convert gallons to ounces by multiplying by several conversion factors. Refer to the table on Imperial Systems of Measurement.

	1 gallon
Multiply the measurement to be converted by 1.
Use conversion factors to get to the right unit. Simplify.
Multiply.
Simplify.	There are 128 ounces in a gallon.

Use Mixed Units of Measurement in the Imperial System

We often use mixed units of measurement in everyday situations. Suppose Joe is 5 feet 10 inches tall, stays at work for 7 hours and 45 minutes, and then eats a 1 pound 2 ounce steak for dinner—all these measurements have mixed units.

Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units!

EXAMPLE 5

Seymour bought three steaks for a barbecue. Their weights were 14 ounces; 1 pound, 2 ounces; and 1 pound, 6 ounces. How many total pounds of steak did he buy?

Solution

We will add the weights of the steaks to find the total weight of the steaks.

Add the ounces. Then add the pounds.
Convert 22 ounces to pounds and ounces.	1 pound, 6 ounces
Add the pounds and ounces.	2 pounds + 1 pound + 6 ounces
Answer	Seymour bought 3 pounds 6 ounces of steak.

EXAMPLE 6

Anthony bought four planks of wood that were each 6 feet 4 inches long. What is the total length of the wood he purchased?

Solution

We will multiply the length of one plank to find the total length.

Multiply the inches and then the feet.
Convert the 16 inches to feet. Add the feet.
	Anthony bought 25 feet and 4 inches of wood.

Make Unit Conversions in the Metric System

In the metric system, units are related by powers of 10. The roots words of their names reflect this relation. For example, the basic unit for measuring length is a metre. One kilometre is 1,000 metres; the prefix kilo means thousand. One centimetre is of a metre, just like one cent is of one dollar.

The equivalencies of measurements in the metric system are shown in the table below. The common abbreviations for each measurement are given in parentheses.

To make conversions in the metric system, we will use the same technique we did in the Imperial system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.

Have you ever run a 5K or 10K race? The length of those races are measured in kilometres. The metric system is commonly used in Canada when talking about the length of a race.

EXAMPLE 7

Nick ran a 10K race. How many metres did he run?

Solution

We will convert kilometres to metres using the identity property of multiplication.

	10 kilometres
Multiply the measurement to be converted by 1.	10 kilometres × 1
Write 1 as a fraction relating kilometres and metres.
Simplify.
Multiply.	10,000 metres
	Nick ran 10,000 metres.

EXAMPLE 8

Eleanor’s newborn baby weighed 3,200 grams. How many kilograms did the baby weigh?

Solution

We will convert grams into kilograms.

Multiply the measurement to be converted by 1.
Write 1 as a function relating kilograms and grams.
Simplify.
Multiply.
Divide.	3.2 kilograms The baby weighed 3.2 kilograms.

TRY IT 8.1

Kari’s newborn baby weighed 2,800 grams. How many kilograms did the baby weigh?

Show answer

2.8 kilograms

As you become familiar with the metric system you may see a pattern. Since the system is based on multiples of ten, the calculations involve multiplying by multiples of ten. We have learned how to simplify these calculations by just moving the decimal.

To multiply by 10, 100, or 1,000, we move the decimal to the right one, two, or three places, respectively. To multiply by 0.1, 0.01, or 0.001, we move the decimal to the left one, two, or three places, respectively.

We can apply this pattern when we make measurement conversions in the metric system. In Example 8, we changed 3,200 grams to kilograms by multiplying by (or 0.001). This is the same as moving the decimal three places to the left.

Figure.1

Use Mixed Units of Measurement in the Imperial System

Performing arithmetic operations on measurements with mixed units of measures in the imperial system requires the same care we used in the Canadian system. Make sure to add or subtract like units.

EXAMPLE 10

Ryland is 1.6 metres tall. His younger brother is 85 centimetres tall. How much taller is Ryland than his younger brother?

Solution

We can convert both measurements to either centimetres or metres. Since metres is the larger unit, we will subtract the lengths in metres. We convert 85 centimetres to metres by moving the decimal 2 places to the left.

Write the 85 centimetres as metres.

Ryland is 0.75 m taller than his brother.

EXAMPLE 11

Dena’s recipe for lentil soup calls for 150 millilitre s of olive oil. Dena wants to triple the recipe. How many litres of olive oil will she need?

Solution

We will find the amount of olive oil in millileters then convert to litres.

	Triple 150 mL
Translate to algebra.
Multiply.	450 mL
Convert to litres.
Simplify.	0.45 L
	Dena needs 0.45 litres of olive oil.

TRY IT 11.1

A recipe for Alfredo sauce calls for 250 millilitre s of milk. Renata is making pasta with Alfredo sauce for a big party and needs to multiply the recipe amounts by 8. How many litres of milk will she need?

Show answer

2 litres

Convert Between the Imperial and the Metric Systems of Measurement

Many measurements in Canada are made in metric units. Our soda may come in 2-litre bottles, our calcium may come in 500-mg capsules, and we may run a 5K race. To work easily in both systems, we need to be able to convert between the two systems.

The table below shows some of the most common conversions.

Conversion Factors Between Imperial and Metric Systems

Length	Mass	Capacity

(Figure.2) shows how inches and centimetres are related on a ruler.

Figure.2

(Figure.3) shows the ounce and millilitre markings on a measuring cup.

Figure.3

(Figure.4) shows how pounds and kilograms marked on a bathroom scale.

Figure.4

We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors.

EXAMPLE 12

Lee’s water bottle holds 500 mL of water. How many ounces are in the bottle? Round to the nearest tenth of an ounce.

Solution

	500 mL
Multiply by a unit conversion factor relating mL and ounces.
Simplify.
Divide.	16.7 ounces.
	The water bottle has 16.7 ounces.

TRY IT 12.1

How many quarts of soda are in a 2-L bottle?

Show answer

2.12 quarts

TRY IT 12.2

How many litres are in 4 quarts of milk?

Show answer

3.8 litres

EXAMPLE 13

Soleil was on a road trip and saw a sign that said the next rest stop was in 100 kilometres. How many miles until the next rest stop?

Solution

	100 kilometres
Multiply by a unit conversion factor relating km and mi.
Simplify.
Divide.	62 ounces.
	Soleil will travel 62 miles.

TRY IT 13.1

The height of Mount Kilimanjaro is 5,895 metres. Convert the height to feet.

Show answer

19,335.6 feet

TRY IT 13.2

The flight distance from Toronto to Vancouver is 3,364 kilometres. Convert the distance to miles.

Show answer

2,090 miles

Convert between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for °F what does that mean?

The Canadian and imperial systems use different scales to measure temperature. The Canadian system uses degrees Celsius, written °C. The imperial system uses degrees Fahrenheit, written°F. (Figure.5) shows the relationship between the two systems.

The diagram shows normal body temperature, along with the freezing and boiling temperatures of water in degrees Fahrenheit and degrees Celsius.

Figure.5

Temperature Conversion

To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula

.

To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula

.

EXAMPLE 14

Convert 50° Fahrenheit into degrees Celsius.

Solution

We will substitute 50°F into the formula to find C.

Simplify in parentheses.
Multiply.
	So we found that 50°F is equivalent to 10°C.

TRY IT 14.1

Convert the Fahrenheit temperature to degrees Celsius: 59° Fahrenheit.

Show answer

15°C

TRY IT 14.2

Convert the Fahrenheit temperature to degrees Celsius: 41° Fahrenheit.

Show answer

5°C

EXAMPLE 15

While visiting Paris, Woody saw the temperature was 20° Celsius. Convert the temperature into degrees Fahrenheit.

Solution

We will substitute 20°C into the formula to find F.

Multiply.
Add.
	So we found that 20°C is equivalent to 68°F.

TRY IT 15.1

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Helsinki, Finland, was 15° Celsius.

Show answer

59°F

TRY IT 15.2

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Sydney, Australia, was 10° Celsius.

Show answer

50° F

Key Concepts

• Metric System of Measurement
  • Length
    
  • Mass
    
  • Capacity
    
• Temperature Conversion
  • To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula
  • To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula

Practice Makes Perfect

Make Unit Conversions in the Imperial System

In the following exercises, convert the units.

1. A park bench is 6 feet long. Convert the length to inches.	2. A floor tile is 2 feet wide. Convert the width to inches.
3. A ribbon is 18 inches long. Convert the length to feet.	4. Carson is 45 inches tall. Convert his height to feet.
5. A football field is 160 feet wide. Convert the width to yards.	6. On a baseball diamond, the distance from home plate to first base is 30 yards. Convert the distance to feet.
7. Ulises lives 1.5 miles from school. Convert the distance to feet.	8. Denver, Colorado, is 5,183 feet above sea level. Convert the height to miles.
9. A killer whale weighs 4.6 tons. Convert the weight to pounds.	10. Blue whales can weigh as much as 150 tons. Convert the weight to pounds.
11. An empty bus weighs 35,000 pounds. Convert the weight to tons.	12. At take-off, an airplane weighs 220,000 pounds. Convert the weight to tons.
13. Rocco waited hours for his appointment. Convert the time to seconds.	14. Misty’s surgery lasted hours. Convert the time to seconds.
15. How many teaspoons are in a pint?	16. How many tablespoons are in a gallon?
17. JJ’s cat, Posy, weighs 14 pounds. Convert her weight to ounces.	18. April’s dog, Beans, weighs 8 pounds. Convert his weight to ounces.
19. Crista will serve 20 cups of juice at her son’s party. Convert the volume to gallons.	20. Lance needs 50 cups of water for the runners in a race. Convert the volume to gallons.
21. Jon is 6 feet 4 inches tall. Convert his height to inches.	22. Faye is 4 feet 10 inches tall. Convert her height to inches.
23. The voyage of the Mayflower took 2 months and 5 days. Convert the time to days.	24. Lynn’s cruise lasted 6 days and 18 hours. Convert the time to hours.
25. Baby Preston weighed 7 pounds 3 ounces at birth. Convert his weight to ounces.	26. Baby Audrey weighted 6 pounds 15 ounces at birth. Convert her weight to ounces.

Use Mixed Units of Measurement in the Imperial System

In the following exercises, solve.

Make Unit Conversions in the Metric System

In the following exercises, convert the units.

Use Mixed Units of Measurement in the Metric System

In the following exercises, solve.

Convert Between the Imperial and the Metric Systems of Measurement

In the following exercises, make the unit conversions. Round to the nearest tenth.

Convert between Fahrenheit and Celsius Temperatures

In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.

In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

Everyday Math

Writing Exercises

Answers

1. 72 inches	3. 1.5 feet	5. yards
7. 7,920 feet	9. 9,200 pounds	11. tons
13. 5,400 s	15. teaspoons	17. ounces
19. gallons	21. 76 in.	23. 65 days
25. 115 ounces	27. 8 lbs. 13 oz.	29. 3.05 hours
31. 10 ft. 2 in.	33. 4 yards	35. 5,000 metres
37. 155 centimetres	39. 3.072 kilometres	41. 1.5 grams
43. 91,600 milligrams	45. 2,000 grams	47. 0.75 litres
49. 91 centimetres	49. 91 centimetres	49. 91 centimetres
53. 2.1 kilograms	55. 1.6 litres	57. 190.5 centimetres
59. 21.9 metres	61. 985.5 kilograms	63. 3.1 miles
65. 44 pounds	67. 53.2 litres	69. 30°C
71. 40°C	73. 22.2°C	75. -17.8°C
77. 41°F	79. 14°F	81. 71.6°F
83. 109.4°F	85. 14.6 kilograms	87. Answers may vary.

Attributions

This chapter has been adapted from “Systems of Measurement” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

Understand Linear, Square, and Cubic Measure

When you measure your height or the length of a garden hose, you use a ruler or tape measure (Figure.1). A tape measure might remind you of a line—you use it for linear measure, which measures length. Inch, foot, yard, mile, centimetre and metre are units of linear measure.

This tape measure measures inches along the top and centimetres along the bottom.

Figure.1

When you want to know how much tile is needed to cover a floor, or the size of a wall to be painted, you need to know the area, a measure of the region needed to cover a surface. Area is measured is square units. We often use square inches, square feet, square centimetres, or square miles to measure area. A square centimetre is a square that is one centimetre (cm) on each side. A square inch is a square that is one inch on each side (Figure.2).

Square measures have sides that are each unit in length.

Figure.2

(Figure.3) shows a rectangular rug that is feet long by feet wide. Each square is foot wide by foot long, or square foot. The rug is made of squares. The area of the rug is square feet.

Figure 3 The rug contains six squares of 1 square foot each, so the total area of the rug is 6 square feet.

When you measure how much it takes to fill a container, such as the amount of gasoline that can fit in a tank, or the amount of medicine in a syringe, you are measuring volume. Volume is measured in cubic units such as cubic inches or cubic centimetres. When measuring the volume of a rectangular solid, you measure how many cubes fill the container. We often use cubic centimetres, cubic inches, and cubic feet. A cubic centimetre is a cube that measures one centimetre on each side, while a cubic inch is a cube that measures one inch on each side (Figure.4).

Figure 4 Cubic measures have sides that are 1 unit in length.

Suppose the cube in (Figure.5) measures inches on each side and is cut on the lines shown. How many little cubes does it contain? If we were to take the big cube apart, we would find little cubes, with each one measuring one inch on all sides. So each little cube has a volume of cubic inch, and the volume of the big cube is cubic inches.

A cube that measures 3 inches on each side is made up of 27 one-inch cubes, or 27 cubic inches.

Figure.5

EXAMPLE 1

For each item, state whether you would use linear, square, or cubic measure:

a) amount of carpeting needed in a room

b) extension cord length

c) amount of sand in a sandbox

d) length of a curtain rod

e) amount of flour in a canister

f) size of the roof of a doghouse.

Solution

a) You are measuring how much surface the carpet covers, which is the area.	square measure
b) You are measuring how long the extension cord is, which is the length.	linear measure
c) You are measuring the volume of the sand.	cubic measure
d) You are measuring the length of the curtain rod.	linear measure
e) You are measuring the volume of the flour.	cubic measure
f) You are measuring the area of the roof.	square measure

TRY IT 1.1

Determine whether you would use linear, square, or cubic measure for each item.

a) amount of paint in a can b) height of a tree c) floor of your bedroom d) diametre of bike wheel e) size of a piece of sod f) amount of water in a swimming pool

Show answer

1. cubic
2. linear
3. square
4. linear
5. square
6. cubic

TRY IT 1.2

Determine whether you would use linear, square, or cubic measure for each item.

a) volume of a packing box b) size of patio c) amount of medicine in a syringe d) length of a piece of yarn e) size of housing lot f) height of a flagpole

Show answer

1. cubic
2. square
3. cubic
4. linear
5. square
6. linear

Many geometry applications will involve finding the perimeter or the area of a figure. There are also many applications of perimeter and area in everyday life, so it is important to make sure you understand what they each mean.

Picture a room that needs new floor tiles. The tiles come in squares that are a foot on each side—one square foot. How many of those squares are needed to cover the floor? This is the area of the floor.

Next, think about putting new baseboard around the room, once the tiles have been laid. To figure out how many strips are needed, you must know the distance around the room. You would use a tape measure to measure the number of feet around the room. This distance is the perimeter.

Perimeter and Area

The perimeter is a measure of the distance around a figure.

The area is a measure of the surface covered by a figure.

(Figure. 6) shows a square tile that is inch on each side. If an ant walked around the edge of the tile, it would walk inches. This distance is the perimeter of the tile.

Since the tile is a square that is inch on each side, its area is one square inch. The area of a shape is measured by determining how many square units cover the shape.

Figure 6 When the ant walks completely around the tile on its edge, it is tracing the perimeter of the tile. The area of the tile is 1 square inch.

EXAMPLE 2

Each of two square tiles is square inch. Two tiles are shown together.

a) What is the perimeter of the figure?

b) What is the area?

Solution

a) The perimeter is the distance around the figure. The perimeter is inches.

b) The area is the surface covered by the figure. There are square inch tiles so the area is square inches.

 

TRY IT 2.1

Find the a) perimeter and b) area of the figure:

Show answer

1. 8 inches
2. 3 sq. inches

TRY IT 2.2

Find the a) perimeter and b) area of the figure:

Show answer

1. 8 centimetres
2. 4 sq. centimetres

Use the Properties of Rectangles

A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, , and the adjacent side as the width, . See (Figure.7).

A rectangle has four sides, and four right angles. The sides are labeled L for length and W for width.

Figure.7

The perimeter, , of the rectangle is the distance around the rectangle. If you started at one corner and walked around the rectangle, you would walk units, or two lengths and two widths. The perimeter then is

What about the area of a rectangle? Remember the rectangular rug from the beginning of this section. It was feet long by feet wide, and its area was square feet. See (Figure.8). Since , we see that the area, , is the length, , times the width, , so the area of a rectangle is .

The area of this rectangular rug is square feet, its length times its width.

Figure.8

Properties of Rectangles

• Rectangles have four sides and four right ° angles.
• The lengths of opposite sides are equal.
• The perimeter, , of a rectangle is the sum of twice the length and twice the width. See (Figure 8).

• The area, , of a rectangle is the length times the width.

For easy reference as we work the examples in this section, we will state the Problem Solving Strategy for Geometry Applications here.

EXAMPLE 3

The length of a rectangle is metres and the width is metres. Find a) the perimeter, and b) the area.

Solution

a)
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the perimeter of a rectangle
Step 3. Name. Choose a variable to represent it.	Let P = the perimeter
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The perimeter of the rectangle is 104 metres.

b)
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of a rectangle
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The area of the rectangle is 60 square metres.

TRY IT 3.1

The length of a rectangle is yards and the width is yards. Find a) the perimeter and b) the area.

Show answer

1. 340 yd
2. 6000 sq. yd

TRY IT 3.2

The length of a rectangle is feet and the width is feet. Find a) the perimeter and b) the area.

Show answer

1. 220 ft
2. 2976 sq. ft

EXAMPLE 4

Find the length of a rectangle with perimeter inches and width inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the length of the rectangle
Step 3. Name. Choose a variable to represent it.	Let L = the length
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length is 15 inches.

TRY IT 4.1

Find the length of a rectangle with a perimeter of inches and width of inches.

Show answer

15 in.

In the next example, the width is defined in terms of the length. We’ll wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

EXAMPLE 5

The width of a rectangle is two inches less than the length. The perimeter is inches. Find the length and width.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length and width of the rectangle
Step 3. Name. Choose a variable to represent it. Now we can draw a figure using these expressions for the length and width.	Since the width is defined in terms of the length, we let L = length. The width is two feet less that the length, so we let L − 2 = width
Step 4.Translate. Write the appropriate formula. The formula for the perimeter of a rectangle relates all the information. Substitute in the given information.
Step 5. Solve the equation.
Combine like terms.
Add 4 to each side.
Divide by 4.
	The length is 14 inches.
Now we need to find the width.
The width is L − 2.	The width is 12 inches.
Step 6. Check: Since , this works!
Step 7. Answer the question.	The length is 14 feet and the width is 12 feet.

TRY IT 5.2

The length of a rectangle is eight feet more than the width. The perimeter is feet. Find the length and width.

Show answer

11 ft , 19 ft

EXAMPLE 6

The length of a rectangle is four centimetres more than twice the width. The perimeter is centimetres. Find the length and width.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length and width
Step 3. Name. Choose a variable to represent it.	let W = width The length is four more than twice the width. 2w + 4 = length
Step 4.Translate. Write the appropriate formula and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length is 12 cm and the width is 4 cm.

TRY IT 6.1

The length of a rectangle is eight more than twice the width. The perimeter is feet. Find the length and width.

Show answer

8 ft, 24 ft

EXAMPLE 7

The area of a rectangular room is square feet. The length is feet. What is the width?

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the width of a rectangular room
Step 3. Name. Choose a variable to represent it.	Let W = width
Step 4.Translate. Write the appropriate formula and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The width of the room is 12 feet.

TRY IT 7.1

The area of a rectangle is square feet. The length is feet. What is the width?

Show answer

26 ft

TRY IT 7.2

The width of a rectangle is metres. The area is square metres. What is the length?

Show answer

29 m

EXAMPLE 8

The perimeter of a rectangular swimming pool is feet. The length is feet more than the width. Find the length and width.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the length and width of the pool
Step 3. Name. Choose a variable to represent it. The length is 15 feet more than the width.	Let
Step 4.Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length of the pool is 45 feet and the width is 30 feet.

Use the Properties of Triangles

We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle in (Figure.9), we’ve labeled the length and the width , so it’s area is .

The area of a rectangle is the base, , times the height, .

Figure.9

We can divide this rectangle into two congruent triangles (Figure.10). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or . This example helps us see why the formula for the area of a triangle is .

A rectangle can be divided into two triangles of equal area. The area of each triangle is one-half the area of the rectangle.

Figure.10

The formula for the area of a triangle is , where is the base and is the height.

To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a ° angle with the base. (Figure.11) shows three triangles with the base and height of each marked.

The height of a triangle is the length of a line segment that connects the the base to the opposite vertex and makes a ° angle with the base.

Figure.11

Triangle Properties

For any triangle , the sum of the measures of the angles is °.

°

The perimeter of a triangle is the sum of the lengths of the sides.

The area of a triangle is one-half the base, , times the height, .

EXAMPLE 9

Find the area of a triangle whose base is inches and whose height is inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the triangle
Step 3. Name. Choose a variable to represent it.	let A = area of the triangle
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The area is 44 square inches.

TRY IT 9.1

Find the area of a triangle with base inches and height inches.

Show answer

13 sq. in.

TRY IT 9.2

Find the area of a triangle with base inches and height inches.

Show answer

49 sq. in.

EXAMPLE 10

The perimeter of a triangular garden is feet. The lengths of two sides are feet and feet. How long is the third side?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	length of the third side of a triangle
Step 3. Name. Choose a variable to represent it.	Let c = the third side
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The third side is 11 feet long.

TRY IT 10.1

The perimeter of a triangular garden is feet. The lengths of two sides are feet and feet. How long is the third side?

Show answer

8 ft

TRY IT 10.2

The lengths of two sides of a triangular window are feet and feet. The perimeter is feet. How long is the third side?

Show answer

6 ft

EXAMPLE 11

The area of a triangular church window is square metres. The base of the window is metres. What is the window’s height?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	height of a triangle
Step 3. Name. Choose a variable to represent it.	Let h = the height
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The height of the triangle is 12 metres.

TRY IT 11.1

The area of a triangular painting is square inches. The base is inches. What is the height?

Show answer

14 in.

TRY IT 11.2

A triangular tent door has an area of square feet. The height is feet. What is the base?

Show answer

6 ft

Isosceles and Equilateral Triangles

Besides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an isosceles triangle. A triangle that has three sides of equal length is called an equilateral triangle. (Figure.12) shows both types of triangles.

In an isosceles triangle, two sides have the same length, and the third side is the base. In an equilateral triangle, all three sides have the same length.

Figure.12

Isosceles and Equilateral Triangles

An isosceles triangle has two sides the same length.

An equilateral triangle has three sides of equal length.

EXAMPLE 12

The perimeter of an equilateral triangle is inches. Find the length of each side.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	Perimeter = 93 in.
Step 2. Identify what you are looking for.	length of the sides of an equilateral triangle
Step 3. Name. Choose a variable to represent it.	Let s = length of each side
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	Each side is 31 inches

TRY IT 12.1

Find the length of each side of an equilateral triangle with perimeter inches.

Show answer

13 in.

TRY IT 12.2

Find the length of each side of an equilateral triangle with perimeter centimetres.

Show answer

17 cm

EXAMPLE 13

Arianna has inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of inches. How long can she make the two equal sides?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	P = 156 in.
Step 2. Identify what you are looking for.	the lengths of the two equal sides
Step 3. Name. Choose a variable to represent it.	Let s = the length of each side
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	Arianna can make each of the two equal sides 48 inches l

TRY IT 13.1

A backyard deck is in the shape of an isosceles triangle with a base of feet. The perimeter of the deck is feet. How long is each of the equal sides of the deck?

Show answer

14 ft

TRY IT 13.2

A boat’s sail is an isosceles triangle with base of metres. The perimeter is metres. How long is each of the equal sides of the sail?

Show answer

7 m

Use the Properties of Trapezoids

A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base , and the length of the bigger base . The height, , of a trapezoid is the distance between the two bases as shown in (Figure.13).

A trapezoid has a larger base, , and a smaller base, . The height is the distance between the bases.

Figure.13

Formula for the Area of a Trapezoid

Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See (Figure.14).

Splitting a trapezoid into two triangles may help you understand the formula for its area.

Figure.14

The height of the trapezoid is also the height of each of the two triangles. See (Figure.15).

Figure.15

The formula for the area of a trapezoid is

If we distribute, we get,

Properties of Trapezoids

• A trapezoid has four sides. See (Figure.13).
• Two of its sides are parallel and two sides are not.
• The area, , of a trapezoid is .

EXAMPLE 14

Find the area of a trapezoid whose height is 6 inches and whose bases are and inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the trapezoid
Step 3. Name. Choose a variable to represent it.	Let
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Is this answer reasonable?

If we draw a rectangle around the trapezoid that has the same big base and a height , its area should be greater than that of the trapezoid.

If we draw a rectangle inside the trapezoid that has the same little base and a height , its area should be smaller than that of the trapezoid.

The area of the larger rectangle is square inches and the area of the smaller rectangle is square inches. So it makes sense that the area of the trapezoid is between and square inches

Step 7. Answer the question. The area of the trapezoid is square inches.

TRY IT 14.1

The height of a trapezoid is yards and the bases are and yards. What is the area?

Show answer

161 sq. yd

TRY IT 14.2

The height of a trapezoid is centimetres and the bases are and centimetres. What is the area?

Show answer

225 sq. cm

EXAMPLE 15

Find the area of a trapezoid whose height is feet and whose bases are and feet.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the trapezoid
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Is this answer reasonable? The area of the trapezoid should be less than the area of a rectangle with base 13.7 and height 5, but more than the area of a rectangle with base 10.3 and height 5.
Step 7. Answer the question.	The area of the trapezoid is 60 square feet.

TRY IT 15.1

The height of a trapezoid is centimetres and the bases are and centimetres. What is the area?

Show answer

42 sq. cm

TRY IT 15.2

The height of a trapezoid is metres and the bases are and metres. What is the area?

Show answer

63 sq. m

EXAMPLE 16

Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of yards and the bases are and yards. How many square yards will be available to plant?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of a trapezoid
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Is this answer reasonable? Yes. The area of the trapezoid is less than the area of a rectangle with a base of 8.2 yd and height 3.4 yd, but more than the area of a rectangle with base 5.6 yd and height 3.4 yd.
Step 7. Answer the question.	Vinny has 23.46 square yards in which he can plan

TRY IT 16.1

Lin wants to sod his lawn, which is shaped like a trapezoid. The bases are yards and yards, and the height is yards. How many square yards of sod does he need?

Show answer

40.25 sq. yd

TRY IT 16.2

Kira wants cover his patio with concrete pavers. If the patio is shaped like a trapezoid whose bases are feet and feet and whose height is feet, how many square feet of pavers will he need?

Show answer

240 sq. ft.

Access Additional Online Resources

• Perimeter of a Rectangle
• Area of a Rectangle
• Perimeter and Area Formulas
• Area of a Triangle
• Area of a Triangle with Fractions
• Area of a Trapezoid

Key Concepts

• Properties of Rectangles
  • Rectangles have four sides and four right (90°) angles.
  • The lengths of opposite sides are equal.
  • The perimeter, , of a rectangle is the sum of twice the length and twice the width.
    • 
  • The area, , of a rectangle is the length times the width.
    • 
• Triangle Properties
  • For any triangle , the sum of the measures of the angles is 180°.
    • °
  • The perimeter of a triangle is the sum of the lengths of the sides.
    • 
  • The area of a triangle is one-half the base, b, times the height, h.
    • 

Glossary

Practice Makes Perfect

Understand Linear, Square, and Cubic Measure

In the following exercises, determine whether you would measure each item using linear, square, or cubic units.

In the following exercises, find the a) perimeter and b) area of each figure. Assume each side of the square is cm.

7.	8.
9.	10.
11.	12.

Use the Properties of Rectangles

In the following exercises, find the a) perimeter and b) area of each rectangle.

13. The length of a rectangle is feet and the width is feet.	14. The length of a rectangle is inches and the width is inches.
15. A rectangular room is feet wide by feet long.	16. A driveway is in the shape of a rectangle feet wide by feet long.

In the following exercises, solve.

17. Find the length of a rectangle with perimeter inches and width inches.	18. Find the length of a rectangle with perimeter yards and width of yards.
19. Find the width of a rectangle with perimeter metres and length metres.	20. Find the width of a rectangle with perimeter metres and length metres.
21. The area of a rectangle is square metres. The length is metres. What is the width?	22. The area of a rectangle is square centimetres. The width is centimetres. What is the length?
23. The length of a rectangle is inches more than the width. The perimeter is inches. Find the length and the width.	24. The width of a rectangle is inches more than the length. The perimeter is inches. Find the length and the width.
25. The perimeter of a rectangle is metres. The width of the rectangle is metres less than the length. Find the length and the width of the rectangle.	26. The perimeter of a rectangle is feet. The width is feet less than the length. Find the length and the width.
27. The width of the rectangle is metres less than the length. The perimeter of a rectangle is metres. Find the dimensions of the rectangle.	28. The length of the rectangle is metres less than the width. The perimeter of a rectangle is metres. Find the dimensions of the rectangle.
29. The perimeter of a rectangle of feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.	30. The length of a rectangle is three times the width. The perimeter is feet. Find the length and width of the rectangle.
31. The length of a rectangle is metres less than twice the width. The perimeter is metres. Find the length and width.	32. The length of a rectangle is inches more than twice the width. The perimeter is inches. Find the length and width.
33. The width of a rectangular window is inches. The area is square inches. What is the length?	34. The length of a rectangular poster is inches. The area is square inches. What is the width?
35. The area of a rectangular roof is square metres. The length is metres. What is the width?	36. The area of a rectangular tarp is square feet. The width is feet. What is the length?
37. The perimeter of a rectangular courtyard is feet. The length is feet more than the width. Find the length and the width.	38. The perimeter of a rectangular painting is centimetres. The length is centimetres more than the width. Find the length and the width.
39. The width of a rectangular window is inches less than the height. The perimeter of the doorway is inches. Find the length and the width.	40. The width of a rectangular playground is metres less than the length. The perimeter of the playground is metres. Find the length and the width.

Use the Properties of Triangles

In the following exercises, solve using the properties of triangles.

41. Find the area of a triangle with base inches and height inches.	42. Find the area of a triangle with base centimetres and height centimetres.
43. Find the area of a triangle with base metres and height metres.	44. Find the area of a triangle with base feet and height feet.
45. A triangular flag has base of foot and height of feet. What is its area?	46. A triangular window has base of feet and height of feet. What is its area?
47. If a triangle has sides of feet and feet and the perimeter is feet, how long is the third side?	48. If a triangle has sides of centimetres and centimetres and the perimeter is centimetres, how long is the third side?
49. What is the base of a triangle with an area of square inches and height of inches?	50. What is the height of a triangle with an area of square inches and base of inches?
51. The perimeter of a triangular reflecting pool is yards. The lengths of two sides are yards and yards. How long is the third side?	52. A triangular courtyard has perimeter of metres. The lengths of two sides are metres and metres. How long is the third side?
53. An isosceles triangle has a base of centimetres. If the perimeter is centimetres, find the length of each of the other sides.	54. An isosceles triangle has a base of inches. If the perimeter is inches, find the length of each of the other sides.
55. Find the length of each side of an equilateral triangle with a perimeter of yards.	56. Find the length of each side of an equilateral triangle with a perimeter of metres.
57. The perimeter of an equilateral triangle is metres. Find the length of each side.	58. The perimeter of an equilateral triangle is miles. Find the length of each side.
59. The perimeter of an isosceles triangle is feet. The length of the shortest side is feet. Find the length of the other two sides.	60. The perimeter of an isosceles triangle is inches. The length of the shortest side is inches. Find the length of the other two sides.
61. A dish is in the shape of an equilateral triangle. Each side is inches long. Find the perimeter.	62. A floor tile is in the shape of an equilateral triangle. Each side is feet long. Find the perimeter.
63. A road sign in the shape of an isosceles triangle has a base of inches. If the perimeter is inches, find the length of each of the other sides.	64. A scarf in the shape of an isosceles triangle has a base of metres. If the perimeter is metres, find the length of each of the other sides.
65. The perimeter of a triangle is feet. One side of the triangle is foot longer than the second side. The third side is feet longer than the second side. Find the length of each side.	66. The perimeter of a triangle is feet. One side of the triangle is feet longer than the second side. The third side is feet longer than the second side. Find the length of each side.
67. One side of a triangle is twice the smallest side. The third side is feet more than the shortest side. The perimeter is feet. Find the lengths of all three sides.	68. One side of a triangle is three times the smallest side. The third side is feet more than the shortest side. The perimeter is feet. Find the lengths of all three sides.

Use the Properties of Trapezoids

In the following exercises, solve using the properties of trapezoids.

69. The height of a trapezoid is feet and the bases are and feet. What is the area?	70. The height of a trapezoid is yards and the bases are and yards. What is the area?
71. Find the area of a trapezoid with a height of metres and bases of and metres.	72. Find the area of a trapezoid with a height of inches and bases of and inches.
73. The height of a trapezoid is centimetres and the bases are and centimetres. What is the area?	74. The height of a trapezoid is feet and the bases are and feet. What is the area?
75. Find the area of a trapezoid with a height of metres and bases of and metres.	76. Find the area of a trapezoid with a height of centimetres and bases of and centimetres.
77. Laurel is making a banner shaped like a trapezoid. The height of the banner is feet and the bases are and feet. What is the area of the banner?	78. Niko wants to tile the floor of his bathroom. The floor is shaped like a trapezoid with width feet and lengths feet and feet. What is the area of the floor?
79. Theresa needs a new top for her kitchen counter. The counter is shaped like a trapezoid with width inches and lengths and inches. What is the area of the counter?	80. Elena is knitting a scarf. The scarf will be shaped like a trapezoid with width inches and lengths inches and inches. What is the area of the scarf?

Everyday Math

81. Fence Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be feet. How long can he make the other side if he wants to use the entire roll of fence?	82. Gardening Lupita wants to fence in her tomato garden. The garden is rectangular and the length is twice the width. It will take feet of fencing to enclose the garden. Find the length and width of her garden.
83. Fence Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are feet, feet, and feet. The fence costs per foot. How much will it cost for Christa to fence in her flowerbed?	84. Painting Caleb wants to paint one wall of his attic. The wall is shaped like a trapezoid with height feet and bases feet and feet. The cost of the painting one square foot of wall is about . About how much will it cost for Caleb to paint the attic wall?

Writing Exercises

Answers

Attributions

This chapter has been adapted from “Use Properties of Rectangles, Triangles, and Trapezoids” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

In this section, we will find the volume and surface area of some three-dimensional figures. Since we will be solving applications, we will once again show our Problem-Solving Strategy for Geometry Applications.

Find Volume and Surface Area of Rectangular Solids

A cheer leading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See Figure.1). The amount of paint needed to cover the outside of each box is the surface area, a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.

This wooden crate is in the shape of a rectangular solid.

Figure.1

Each crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in Figure.2 has length units, width units, and height units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This by by rectangular solid has cubic units.

Figure.2

Altogether there are cubic units. Notice that is the

The volume, , of any rectangular solid is the product of the length, width, and height.

We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, , is equal to

We can substitute for in the volume formula to get another form of the volume formula.

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the rectangular solid we started with. See Figure.3.

Figure.3

To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.

Notice for each of the three faces you see, there is an identical opposite face that does not show.

The surface area of the rectangular solid shown in (Figure.3) is square units.

In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see Figure.4). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

For each face of the rectangular solid facing you, there is another face on the opposite side. There are faces in all.

Figure.4

Volume and Surface Area of a Rectangular Solid

For a rectangular solid with length , width , and height

EXAMPLE 1

For a rectangular solid with length cm, height cm, and width cm, find the a) volume and b) surface area.

Solution

Step 1 is the same for both a) and b), so we will show it just once.

Step 1. Read the problem. Draw the figure and label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the rectangular solid
Step 3. Name. Choose a variable to represent it.	Let = volume
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check We leave it to you to check your calculations.
Step 7. Answer the question.	The surface area is square centimetres.

b)
Step 2. Identify what you are looking for.	the surface area of the solid
Step 3. Name. Choose a variable to represent it.	Let = surface area
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Double-check with a calculator.
Step 7. Answer the question.	The surface area is 1,034 square centimetres.

TRY IT 1.1

Find the a) volume and b) surface area of rectangular solid with the: length feet, width feet, and height feet.

Show answer

1. 792 cu. ft
2. 518 sq. ft

TRY IT 1.2

Find the a) volume and b) surface area of rectangular solid with the: length feet, width feet, and height feet.

Show answer

1. 1,440 cu. ft
2. 792 sq. ft

EXAMPLE 2

A rectangular crate has a length of inches, width of inches, and height of inches. Find its a) volume and b) surface area.

Solution

Step 1 is the same for both a) and b), so we will show it just once.

Step 1. Read the problem. Draw the figure and label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the crate
Step 3. Name. Choose a variable to represent it.	let = volume
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Double check your math.
Step 7. Answer the question.	The volume is 15,000 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the crate
Step 3. Name. Choose a variable to represent it.	let = surface area
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Check it yourself!
Step 7. Answer the question.	The surface area is 3,700 square inches.

TRY IT 2.1

A rectangular box has length feet, width feet, and height feet. Find its a) volume and b) surface area.

Show answer

1. 216 cu. ft
2. 228 sq. ft

TRY IT 2.2

A rectangular suitcase has length inches, width inches, and height inches. Find its a) volume and b) surface area.

Show answer

1. 2,772 cu. in.
2. 1,264 sq. in.

Volume and Surface Area of a Cube

A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

So for a cube, the formulas for volume and surface area are and .

Volume and Surface Area of a Cube

For any cube with sides of length ,

EXAMPLE 3

A cube is inches on each side. Find its a) volume and b) surface area.

Solution

Step 1 is the same for both a) and b), so we will show it just once.

Step 1. Read the problem. Draw the figure and label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the cube
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula.
Step 5. Solve. Substitute and solve.
Step 6. Check: Check your work.
Step 7. Answer the question.	The volume is 15.625 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the cube
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula.
Step 5. Solve. Substitute and solve.
Step 6. Check: The check is left to you.
Step 7. Answer the question.	The surface area is 37.5 square inches.

TRY IT 3.1

For a cube with side 4.5 metres, find the a) volume and b) surface area of the cube.

Show answer

1. 91.125 cu. m
2. 121.5 sq. m

TRY IT 3.2

For a cube with side 7.3 yards, find the a) volume and b) surface area of the cube.

Show answer

1. 389.017 cu. yd.
2. 319.74 sq. yd.

EXAMPLE 4

A notepad cube measures inches on each side. Find its a) volume and b) surface area.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the cube
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula.
Step 5. Solve the equation.
Step 6. Check: Check that you did the calculations correctly.
Step 7. Answer the question.	The volume is 8 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the cube
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula.
Step 5. Solve the equation.
Step 6. Check: The check is left to you.
Step 7. Answer the question.	The surface area is 24 square inches.

TRY IT 4.1

A packing box is a cube measuring feet on each side. Find its a) volume and b) surface area.

Show answer

1. 64 cu. ft
2. 96 sq. ft

TRY IT 4.2

A packing box is a cube measuring feet on each side. Find its a) volume and b) surface area.

Show answer

1. 64 cu. ft
2. 96 sq. ft

Find the Volume and Surface Area of Spheres

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the centre of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.

Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course. We will approximate with .

Volume and Surface Area of a Sphere

For a sphere with radius

EXAMPLE 5

A sphere has a radius inches. Find its a) volume and b) surface area.

Solution

Step 1 is the same for both a) and b), so we will show it just once.

Step 1. Read the problem. Draw the figure and label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the sphere
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula.
Step 5. Solve.
Step 6. Check: Double-check your math on a calculator.
Step 7. Answer the question.	The volume is approximately 904.32 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the cube
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula.
Step 5. Solve.
Step 6. Check: Double-check your math on a calculator
Step 7. Answer the question.	The surface area is approximately 452.16 square inches.

TRY IT 5.1

Find the a) volume and b) surface area of a sphere with radius 3 centimetres.

Show answer

1. 113.04 cu. cm
2. 113.04 sq. cm

EXAMPLE 6

A globe of Earth is in the shape of a sphere with radius centimetres. Find its a) volume and b) surface area. Round the answer to the nearest hundredth.

Solution

Step 1. Read the problem. Draw a figure with the given information and label it.

a)
Step 2. Identify what you are looking for.	the volume of the sphere
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for )
Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume is approximately 11,488.21 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the sphere
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for )
Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The surface area is approximately 2461.76 square inches.

TRY IT 6.1

A beach ball is in the shape of a sphere with radius of inches. Find its a) volume and b) surface area.

Show answer

1. 3052.08 cu. in.
2. 1017.36 sq. in.

Find the Volume and Surface Area of a Cylinder

If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, , will be perpendicular to the bases.

A cylinder has two circular bases of equal size. The height is the distance between the bases.

Rectangular solids and cylinders are somewhat similar because they both have two bases and a height. The formula for the volume of a rectangular solid, , can also be used to find the volume of a cylinder.

For the rectangular solid, the area of the base, , is the area of the rectangular base, length × width. For a cylinder, the area of the base, , is the area of its circular base, . (Figure.5) compares how the formula is used for rectangular solids and cylinders.

Seeing how a cylinder is similar to a rectangular solid may make it easier to understand the formula for the volume of a cylinder.

Figure.5

To understand the formula for the surface area of a cylinder, think of a can of vegetables. It has three surfaces: the top, the bottom, and the piece that forms the sides of the can. If you carefully cut the label off the side of the can and unroll it, you will see that it is a rectangle. See (Figure.6).

By cutting and unrolling the label of a can of vegetables, we can see that the surface of a cylinder is a rectangle. The length of the rectangle is the circumference of the cylinder’s base, and the width is the height of the cylinder.

Figure.6

The distance around the edge of the can is the circumference of the cylinder’s base it is also the length of the rectangular label. The height of the cylinder is the width of the rectangular label. So the area of the label can be represented as

To find the total surface area of the cylinder, we add the areas of the two circles to the area of the rectangle.

The surface area of a cylinder with radius and height , is

Volume and Surface Area of a Cylinder

For a cylinder with radius and height

EXAMPLE 7

A cylinder has height centimetres and radius centimetres. Find the a) volume and b) surface area.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the cylinder
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for )
Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume is approximately 141.3 cubic inches.

b)
Step 2. Identify what you are looking for.	the surface area of the cylinder
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for )
Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The surface area is approximately 150.72 square inches.

TRY IT 7.1

Find the a) volume and b) surface area of the cylinder with radius 4 cm and height 7cm.

Show answer

1. 351.68 cu. cm
2. 276.32 sq. cm

TRY IT 7.2

Find the a) volume and b) surface area of the cylinder with given radius 2 ft and height 8 ft.

Show answer

1. 100.48 cu. ft
2. 125.6 sq. ft

EXAMPLE 8

Find the a) volume and b) surface area of a can of soda. The radius of the base is centimetres and the height is centimetres. Assume the can is shaped exactly like a cylinder.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.

a)
Step 2. Identify what you are looking for.	the volume of the cylinder
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for )
Step 5. Solve.
Step 6. Check: We leave it to you to check.
Step 7. Answer the question.	The volume is approximately 653.12 cubic centimetres.

b)
Step 2. Identify what you are looking for.	the surface area of the cylinder
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for )
Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The surface area is approximately 427.04 square centimetres.

TRY IT 8.1

Find the a) volume and b) surface area of a can of paint with radius 8 centimetres and height 19 centimetres. Assume the can is shaped exactly like a cylinder.

Show answer

1. 3,818.24 cu. cm
2. 1,356.48 sq. cm

TRY IT 8.2

Find the a) volume and b) surface area of a cylindrical drum with radius 2.7 feet and height 4 feet. Assume the drum is shaped exactly like a cylinder.

Show answer

1. 91.5624 cu. ft
2. 113.6052 sq. ft

Find the Volume of Cones

The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.

In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See (Figure.6).

The height of a cone is the distance between its base and the vertex.

Figure.6

Earlier in this section, we saw that the volume of a cylinder is . We can think of a cone as part of a cylinder. Figure.7 shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.

The volume of a cone is less than the volume of a cylinder with the same base and height.

Figure.7

In fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is

Since the base of a cone is a circle, we can substitute the formula of area of a circle, , for to get the formula for volume of a cone.

In this book, we will only find the volume of a cone, and not its surface area.

Volume of a Cone

For a cone with radius and height .

EXAMPLE 9

Find the volume of a cone with height inches and radius of its base inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the volume of the cone
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for )
Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume is approximately 25.12 cubic inches.

TRY IT 9.1

Find the volume of a cone with height inches and radius inches

Show answer

65.94 cu. in.

TRY IT 9.2

Find the volume of a cone with height centimetres and radius centimetres

Show answer

235.5 cu. cm

EXAMPLE 10

Marty’s favorite gastro pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is inches tall and inches in diametre? Round the answer to the nearest hundredth.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone.
Step 2. Identify what you are looking for.	the volume of the cone
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for , and notice that we were given the distance across the circle, which is its diametre. The radius is 2.5 inches.)
Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume of the wrap is approximately 52.33 cubic inches.

TRY IT 10.1

How many cubic inches of candy will fit in a cone-shaped piñata that is inches long and inches across its base? Round the answer to the nearest hundredth.

Show answer

678.24 cu. in.

TRY IT 10.2

What is the volume of a cone-shaped party hat that is inches tall and inches across at the base? Round the answer to the nearest hundredth.

Show answer

128.2 cu. in.

ACCESS ADDITIONAL ONLINE RESOURCES

• Volume of a Cone

Key Concepts

• Volume and Surface Area of a Rectangular Solid
  • 
  • 
• Volume and Surface Area of a Cube
  • 
  • 
• Volume and Surface Area of a Sphere
  • 
  • 
• Volume and Surface Area of a Cylinder
  • 
  • 
• Volume of a Cone
  • For a cone with radius and height :
    Volume:

Glossary

Practice Makes Perfect

Find Volume and Surface Area of Rectangular Solids

In the following exercises, find a) the volume and b) the surface area of the rectangular solid with the given dimensions.

1. length metres, width metres, height metres	2. length feet, width feet, height feet
3. length yards, width yards, height yards	4. length centimetres, width centimetres, height centimetres

In the following exercises, solve.

5. Moving van A rectangular moving van has length feet, width feet, and height feet. Find its a) volume and b) surface area.	6. Gift box A rectangular gift box has length inches, width inches, and height inches. Find its a) volume and b) surface area.
7. Carton A rectangular carton has length cm, width cm, and height cm. Find its a) volume and b) surface area.	8.Shipping container A rectangular shipping container has length feet, width feet, and height feet. Find its a) volume and b) surface area.

In the following exercises, find a) the volume and b) the surface area of the cube with the given side length.

9. centimetres	10. inches
11. feet	12. metres

In the following exercises, solve.

13. Science center Each side of the cube at the Discovery Science Center in Santa Ana is feet long. Find its a) volume and b) surface area.	14. Museum A cube-shaped museum has sides metres long. Find its a) volume and b) surface area.
15. Base of statue The base of a statue is a cube with sides metres long. Find its a) volume and b) surface area.	16. Tissue box A box of tissues is a cube with sides 4.5 inches long. Find its a) volume and b) surface area.

Find the Volume and Surface Area of Spheres

In the following exercises, find a) the volume and b) the surface area of the sphere with the given radius. Round answers to the nearest hundredth.

17. centimetres	18. inches
19. feet	20. yards

In the following exercises, solve. Round answers to the nearest hundredth.

21. Exercise ball An exercise ball has a radius of inches. Find its a) volume and b) surface area.	22. Balloon ride The Great Park Balloon is a big orange sphere with a radius of feet . Find its a) volume and b) surface area.
23. Golf ball A golf ball has a radius of centimetres. Find its a) volume and b) surface area.	24. Baseball A baseball has a radius of inches. Find its a) volume and b) surface area.

Find the Volume and Surface Area of a Cylinder

In the following exercises, find a) the volume and b) the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth.

25. radius feet, height feet	26. radius centimetres, height centimetres
27. radius metres, height metres	28. radius yards, height yards

In the following exercises, solve. Round answers to the nearest hundredth.

29. Coffee can A can of coffee has a radius of cm and a height of cm. Find its a) volume and b) surface area.	30. Snack pack A snack pack of cookies is shaped like a cylinder with radius cm and height cm. Find its a) volume and b) surface area.
31. Barber shop pole A cylindrical barber shop pole has a diametre of inches and height of inches. Find its a) volume and b) surface area.	32. Architecture A cylindrical column has a diametre of feet and a height of feet. Find its a) volume and b) surface area.

Find the Volume of Cones

In the following exercises, find the volume of the cone with the given dimensions. Round answers to the nearest hundredth.

33. height feet and radius feet	34. height inches and radius inches
35. height centimetres and radius cm	36. height metres and radius metres

In the following exercises, solve. Round answers to the nearest hundredth.

37. Teepee What is the volume of a cone-shaped teepee tent that is feet tall and feet across at the base?	38. Popcorn cup What is the volume of a cone-shaped popcorn cup that is inches tall and inches across at the base?
39. Silo What is the volume of a cone-shaped silo that is feet tall and feet across at the base?	40. Sand pile What is the volume of a cone-shaped pile of sand that is metres tall and metres across at the base?

Everyday Math

Writing Exercises

43. The formulas for the volume of a cylinder and a cone are similar. Explain how you can remember which formula goes with which shape.

44. Which has a larger volume, a cube of sides of feet or a sphere with a diametre of feet? Explain your reasoning.

Answers

Attributions

• This chapter has been adapted from “Solve Geometry Applications: Volume and Surface Area” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

In this section, we’ll continue working with geometry applications. We will add several new formulas to our collection of formulas. To help you as you do the examples and exercises in this section, we will show the Problem Solving Strategy for Geometry Applications here.

Use the Properties of Circles

 We’ll  refer to the properties of circles as we use them to solve applications.

EXAMPLE 1

A circular sandbox has a radius of feet. Find the a) circumference and b) area of the sandbox.

Solution

a)

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the circumference of the circle
Step 3. Name. Choose a variable to represent it.	Let c = circumference of the circle
Step 4. Translate. Write the appropriate formula Substitute
Step 5. Solve the equation.
Step 6. Check. Does this answer make sense? Yes. If we draw a square around the circle, its sides would be 5 ft (twice the radius), so its perimeter would be 20 ft. This is slightly more than the circle’s circumference, 15.7 ft.
Step 7. Answer the question.	The circumference of the sandbox is 15.7 feet.

 

b)

Step 1. Read the problem. Draw the figure and label it with the given information
Step 2. Identify what you are looking for.	the area of the circle
Step 3. Name. Choose a variable to represent it.	Let A = the area of the circle
Step 4. Translate. Write the appropriate formula Substitute
Step 5. Solve the equation.
Step 6. Check. Yes. If we draw a square around the circle, its sides would be 5 ft, as shown in part a). So the area of the square would be 25 sq. ft. This is slightly more than the circle’s area, 19.625 sq. ft.
Step 7. Answer the question.	The area of the circle is 19.625 square feet.

But since the diametre of a circle is two times the radius, we could write the formula for the circumference in terms .

We will use this form of the circumference when we’re given the length of the diametre instead of the radius.

EXAMPLE 2

A circular table has a diametre of four feet. What is the circumference of the table?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the circumference of the table
Step 3. Name. Choose a variable to represent it.	Let c = the circumference of the table
Step 4. Translate. Write the appropriate formula for the situation. Substitute.
Step 5. Solve the equation, using 3.14 for .
Step 6. Check: If we put a square around the circle, its side would be 4. The perimeter would be 16. It makes sense that the circumference of the circle, 12.56, is a little less than 16.
Step 7. Answer the question.	The diametre of the table is 12.56 square feet

EXAMPLE 3

Find the diametre of a circle with a circumference of centimetres.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the diametre of the circle
Step 3. Name. Choose a variable to represent it.	Let d = the diametre of the circle
Step 4. Translate.
Write the formula. Substitute, using 3.14 to approximate .
Step 5. Solve.
Step 6. Check: ✓
Step 7. Answer the question.	The diametre of the circle is approximately 15 centimetres.

TRY IT 3.1

Find the diametre of a circle with circumference of centimetres.

Show answer

30 cm

Find the Area of Irregular Figures

So far, we have found area for rectangles, triangles, trapezoids, and circles. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. But some irregular figures are made up of two or more standard geometric shapes. To find the area of one of these irregular figures, we can split it into figures whose formulas we know and then add the areas of the figures.

Glossary

Practice Makes Perfect

Use the Properties of Circles

In the following exercises, solve using the properties of circles.

1. The lid of a paint bucket is a circle with radius inches. Find the a) circumference and b) area of the lid.	2. An extra-large pizza is a circle with radius inches. Find the a) circumference and b) area of the pizza.
3. A farm sprinkler spreads water in a circle with radius of feet. Find the a) circumference and b) area of the watered circle.	4. A circular rug has radius of feet. Find the a) circumference and b) area of the rug.
5. A reflecting pool is in the shape of a circle with diametre of feet. What is the circumference of the pool?	6. A turntable is a circle with diametre of inches. What is the circumference of the turntable?
7. A circular saw has a diametre of inches. What is the circumference of the saw?	8. A round coin has a diametre of centimetres. What is the circumference of the coin?
9. A barbecue grill is a circle with a diametre of feet. What is the circumference of the grill?	10. The top of a pie tin is a circle with a diametre of inches. What is the circumference of the top?
11. A circle has a circumference of inches. Find the diametre.	12. A circle has a circumference of feet. Find the diametre.
13. A circle has a circumference of metres. Find the diametre.	14. A circle has a circumference of centimetres. Find the diametre.

In the following exercises, find the radius of the circle with given circumference.

15. A circle has a circumference of feet.	16. A circle has a circumference of centimetres.
17. A circle has a circumference of miles.	18. A circle has a circumference of inches.

Find the Area of Irregular Figures

In the following exercises, find the area of the irregular figure. Round your answers to the nearest hundredth.

19.	20.
21.	22.
23.	24.
25.	26.
27.	28.
29.	30.
31.	32.
33.	34.
35.	36.
37.	38.

In the following exercises, solve.

39. A city park covers one block plus parts of four more blocks, as shown. The block is a square with sides feet long, and the triangles are isosceles right triangles. Find the area of the park.	40. A gift box will be made from a rectangular piece of cardboard measuring inches by inches, with squares cut out of the corners of the sides, as shown. The sides of the squares are inches. Find the area of the cardboard after the corners are cut out.
41. Perry needs to put in a new lawn. His lot is a rectangle with a length of feet and a width of feet. The house is rectangular and measures feet by feet. His driveway is rectangular and measures feet by feet, as shown. Find the area of Perry’s lawn.	42. Denise is planning to put a deck in her back yard. The deck will be a by rectangle with a semicircle of diametre feet, as shown below. Find the area of the deck.

Everyday Math

43. Area of a Tabletop Yuki bought a drop-leaf kitchen table. The rectangular part of the table is a by rectangle with a semicircle at each end, as shown. a) Find the area of the table with one leaf up. b) Find the area of the table with both leaves up.

44. Painting Leora wants to paint the nursery in her house. The nursery is an by rectangle, and the ceiling is feet tall. There is a by door on one wall, a by closet door on another wall, and one by window on the third wall. The fourth wall has no doors or windows. If she will only paint the four walls, and not the ceiling or doors, how many square feet will she need to paint?

Writing Exercises

45. Describe two different ways to find the area of this figure, and then show your work to make sure both ways give the same area.

46. A circle has a diametre of feet. Find the area of the circle a) using for b) using for . c) Which calculation to do prefer? Why?

Answers

Attributions

This chapter has been adapted from “Solve Geometry Applications: Circles and Irregular Figures” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

Review Exercises

Systems of Measurement

In the following exercises, convert between Imperial units. Round to the nearest tenth.

1. A picture frame is inches wide. Convert the width to feet.	2. A floral arbor is feet tall. Convert the height to inches.
3. A playground is feet wide. Convert the width to yards.	4. Kelly is feet inches tall. Convert her height to inches.
5. An orca whale in the Salish Sea weighs tons. Convert the weight to pounds.	6. The height of Mount Shasta is feet. Convert the height to miles.
7. How many tablespoons are in a quart?	8. The play lasted hours. Convert the time to minutes.
9. Trinh needs cups of paint for her class art project. Convert the volume to gallons.	10. Naomi’s baby weighed pounds ounces at birth. Convert the weight to ounces.

In the following exercises, solve, and state your answer in mixed units.

11. Every day last week, Pedro recorded the amount of time he spent reading. He read for minutes. How much time, in hours and minutes, did Pedro spend reading?	12. John caught lobsters. The weights of the lobsters were pound ounces, pound ounces, pounds ounces, and pounds ounces. What was the total weight of the lobsters?
13. Dalila wants to make pillow covers. Each cover takes inches of fabric. How many yards and inches of fabric does she need for pillow covers?	14. Fouad is feet inches tall. If he stands on a rung of a ladder feet inches high, how high off the ground is the top of Fouad’s head?

In the following exercises, convert between metric units.

15. Mount Everest is metres tall. Convert the height to kilometres.	16. Donna is metres tall. Convert her height to centimetres.
17. One cup of yogurt contains grams of protein. Convert this to milligrams.	18. One cup of yogurt contains milligrams of calcium. Convert this to grams.
19. A bottle of water contained millilitre s. Convert this to litres.	20. Sergio weighed kilograms at birth. Convert this to grams.

In the following exercises, solve.

21. Selma had a bottle of water. If she drank millilitres, how much water, in millilitres, was left in the bottle?	22. Minh is metres tall. His daughter is centimetres tall. How much taller, in metres, is Minh than his daughter?
23. One ounce of tofu provides grams of protein. How many milligrams of protein are provided by ounces of tofu?	24. One serving of cranberry juice contains grams of sugar. How many kilograms of sugar are in servings of cranberry juice?

In the following exercises, convert between Imperial and metric units. Round to the nearest tenth.

25. A college basketball court is feet long. Convert this length to metres.	26. Majid is inches tall. Convert his height to centimetres.
27. Lucas weighs kilograms. Convert his weight to pounds.	28. Caroline walked kilometres. Convert this length to miles.
29. A box of books weighs pounds. Convert this weight to kilograms.	30. Steve’s car holds litres of gas. Convert this to gallons.

In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.

31. °F	32. °F
33. °F	34. °F

In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

35. °C	36. °C
37. °C	38. °C

Understand Linear, Square, Cubic Measure

In the following exercises, would you measure each item using linear, square, or cubic measure?

39. amount of sand in a sandbag	40. height of a tree
41. size of a patio	42. length of a highway

In the following exercises, find a) the perimeter b) the area of each figure

43.

44.

Use Properties of Rectangles

In the following exercises, find the a) perimeter b) area of each rectangle

45. The length of a rectangle is metres and the width is metres.	46. The length of a rectangle is feet and the width is feet.
47. A sidewalk in front of Kathy’s house is in the shape of a rectangle feet wide by feet long.	48. A rectangular room is feet wide by feet long.

In the following exercises, solve.

49. Find the length of a rectangle with perimeter of centimetres and width of centimetres.	50. Find the width of a rectangle with perimeter and length .
51. The area of a rectangle is square metres. The length is metres. What is the width?	52. The width of a rectangle is centimetres. The area is square centimetres. What is the length?
53. The length of a rectangle is centimetres more than the width. The perimeter is centimetres. Find the length and the width.	54. The width of a rectangle is more than twice the length. The perimeter is inches. Find the length and the width.

Use Properties of Triangles

In the following exercises, solve using the properties of triangles.

55. Find the area of a triangle with base inches and height inches.	56. Find the area of a triangle with base centimetres and height centimetres.
57. A triangular road sign has base inches and height inches. What is its area?	58. If a triangular courtyard has sides feet and feet and the perimeter is feet, how long is the third side?
59. A tile in the shape of an isosceles triangle has a base of inches. If the perimeter is inches, find the length of each of the other sides.	60. Find the length of each side of an equilateral triangle with perimeter of yards.
61. The perimeter of a triangle is feet. One side of the triangle is feet longer than the shortest side. The third side is feet longer than the shortest side. Find the length of each side.	62. One side of a triangle is three times the smallest side. The third side is feet more than the shortest side. The perimeter is feet. Find the lengths of all three sides.

Use Properties of Trapezoids

In the following exercises, solve using the properties of trapezoids.

63. The height of a trapezoid is feet and the bases are and feet. What is the area?	64. The height of a trapezoid is yards and the bases are and yards. What is the area?
65. Find the area of the trapezoid with height metres and bases and metres.	66. A flag is shaped like a trapezoid with height centimetres and the bases are and centimetres. What is the area of the flag?

Use Properties of Circles

In the following exercises, solve using the properties of circles. Round answers to the nearest hundredth.

67. A circular mosaic has radius metres. Find the a) circumference b) area of the mosaic	68. A circular fountain has radius feet. Find the a) circumference b) area of the fountain
69. Find the diametre of a circle with circumference inches.	70. Find the radius of a circle with circumference centimetres

Find the Area of Irregular Figures

In the following exercises, find the area of each shaded region.

71.	72.
73.	74.
75.	76.

Find Volume and Surface Area of Rectangular Solids

In the following exercises, find the a) volume b) surface area of the rectangular solid

77. A rectangular solid with length centimetres, width centimetres, and height centimetres	78. A cube with sides that are feet long
79. A cube of tofu with sides inches	80. A rectangular carton with length inches, width inches, and height inches

Find Volume and Surface Area of Spheres

In the following exercises, find the a) volume b) surface area of the sphere.

81. a sphere with radius yards	82. a sphere with radius metres
83. a baseball with radius inches	84. a soccer ball with radius centimetres

Find Volume and Surface Area of Cylinders

In the following exercises, find the a) volume b) surface area of the cylinder

85. A cylinder with radius yards and height yards	86. A cylinder with diametre inches and height inches
87. A juice can with diametre centimetres and height centimetres	88. A cylindrical pylon with diametre feet and height feet

Find Volume of Cones

In the following exercises, find the volume of the cone.

89. A cone with height metres and radius metre	90. A cone with height feet and radius feet
91. A cone-shaped water cup with diametre inches and height inches	92. A cone-shaped pile of gravel with diametre yards and height yards

Review Answers

1. 3.5 feet	3. 15 yards	5. 9000 pounds
7. 64 tablespoons	9. 1.9 gallons	11. 7 hours 10 minutes
13. 3 yards, 12 inches	15. 8.85 kilometres	17. 13,000 milligrams
19. 0.65 litres	21. 855 millilitre s	23. 10,000 milligrams
25. 25.6 metres	27. 171.6 pounds	29. 11.4 kilograms
31. °C	33. °C	35. °F
37. °F	39. cubic	41. square
43. a) 8 units b) 3 sq. units	45. a) 140 m b) 1176 sq. m	47. a) 98 ft. b) 180 sq. ft.
49. 25 cm	51. 62 m	53. 24.5 in., 12.5 in.
55. 135 sq. in.	57. 600 sq. in.	59. 7 in., 7 in.
61. 17 ft., 20 ft., 22 ft.	63. 100 sq. ft.	65. 675 sq. m
67. a) 18.84 m b) 28.26 sq. m	69. 48 in.	71. 30 sq. units
73. 300 sq. units	75. 199.25 sq. units	77. a) 630 cu. cm b) 496 sq. cm
79. a) 15.625 cu. in. b) 37.5 sq. in.	81. a) 267.95 cu. yd. b) 200.96 sq. yd.	83. a) 12.76 cu. in. b) 26.41 sq. in.
85. a) 75.36 cu. yd. b) 100.48 sq. yd.	87. a) 753.6 cu. cm b) 477.28 sq. cm	89. 5.233 cu. m
91. 4.599 cu. in.

Practice Test

In the following exercises, solve using the appropriate unit conversions.

1. One cup of milk contains milligrams of calcium. Convert this to grams.	2. Azize walked miles. Convert this distance to feet.
3. Janice ran kilometres. Convert this distance to miles. Round to the nearest hundredth of a mile.	4. Larry had phone customer phone calls yesterday. The calls lasted minutes. How much time, in hours and minutes, did Larry spend on the phone?
5. Use the formula to convert °C to degrees	6. Yolie is inches tall. Convert her height to centimetres. Round to the nearest centimetre.
7. A triangular poster has base centimetres and height centimetres. Find the area of the poster.	8. The length of a rectangle is feet more than five times the width. The perimeter is feet. Find the dimensions of the rectangle.
9. A circular pool has diametre inches. What is its circumference? Round to the nearest tenth.	10. A trapezoid has height inches and bases inches and inches. Find the area of the trapezoid.
11. Find the volume of a rectangular room with width feet, length feet, and height feet.	12. Find the area of the shaded region. Round to the nearest tenth.
13. A traffic cone has height centimetres. The radius of the base is centimetres. Find the volume of the cone. Round to the nearest tenth.	14. A coffee can is shaped like a cylinder with height inches and radius inches. Find (a) the surface area and (b) the volume of the can. Round to the nearest tenth.

Practice Test Answers