CHAPTER 3 Measurement, Perimeter, Area, and Volume

# 3.3 Solve Geometry Applications: Volume and Surface Area

Learning Objectives

By the end of this section, you will be able to:

• Find volume and surface area of rectangular solids
• Find volume and surface area of spheres
• Find volume and surface area of cylinders
• Find volume of cone

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.

Problem Solving Strategy for Geometry Applications

1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
2. Identify what you are looking for.
3. Name what you are looking for. Choose a variable to represent that quantity.
4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

# 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

TRY IT 5.2

Find the a) volume and b) surface area of each sphere with a radius of foot

1. 4.19 cu. ft
2. 12.56 sq. ft

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.

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

TRY IT 6.2

A Roman statue depicts Atlas holding a globe with radius of feet. Find the a) volume and b) surface area of the globe.

1. 14.13 cu. ft
2. 28.26 sq. ft

# 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.

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.

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.

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.

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.

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.

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.

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.

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

65.94 cu. in.

TRY IT 9.2

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

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.

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.

128.2 cu. in.

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# 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

cone
A cone is a solid figure with one circular base and a vertex.
cube
A cube is a rectangular solid whose length, width, and height are equal.
cylinder
A cylinder is a solid figure with two parallel circles of the same size at the top and bottom.

# 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

 41. Street light post The post of a street light is shaped like a truncated cone, as shown in the picture below. It is a large cone minus a smaller top cone. The large cone is feet tall with base radius foot. The smaller cone is feet tall with base radius of feet. To the nearest tenth, a) find the volume of the large cone. b) find the volume of the small cone. c) find the volume of the post by subtracting the volume of the small cone from the volume of the large cone. 42. Ice cream cones A regular ice cream cone is 4 inches tall and has a diametre of inches. A waffle cone is inches tall and has a diametre of inches. To the nearest hundredth, a) find the volume of the regular ice cream cone. b) find the volume of the waffle cone. c) how much more ice cream fits in the waffle cone compared to the regular cone?

## 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.