9.7: Solve Geometry Applications- Volume and Surface Area
- Page ID
- 114988
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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 cones
Be Prepared 9.16
Before you get started, take this readiness quiz.
Evaluate when
If you missed this problem, review Example 2.15.
Be Prepared 9.17
Evaluate when
If you missed this problem, review Example 2.16.
Be Prepared 9.18
Find the area of a circle with radius
If you missed this problem, review Example 5.39.
In this section, we will finish our study of geometry applications. We 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.
- Step 1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
- Step 2. Identify what you are looking for.
- Step 3. Name what you are looking for. Choose a variable to represent that quantity.
- Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
- Step 5. Solve the equation using good algebra techniques.
- Step 6. Check the answer in the problem and make sure it makes sense.
- Step 7. Answer the question with a complete sentence.
Find Volume and Surface Area of Rectangular Solids
A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See Figure 9.28). 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.
Each crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in Figure 9.29 has length units, width units, and height units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.
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
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
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 two dimensions, either length and width, length and height, or width and height (see Figure 9.31). 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.
Volume and Surface Area of a Rectangular Solid
For a rectangular solid with length
Manipulative Mathematics
Example 9.47
For a rectangular solid with length
- Answer
-
Step 1 is the same for both ⓐ and ⓑ, so we will show it just once.
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 rectangular solid Step 3. Name. Choose a variable to represent it. Let = volumeV V Step 4. Translate.
Write the appropriate formula.
Substitute.
V = L W H V = L W H
V = 14 ⋅ 9 ⋅ 17 V = 14 ⋅ 9 ⋅ 17 Step 5. Solve the equation. V = 2, 142 V = 2,142 Step 6. Check
We leave it to you to check your calculations.Step 7. Answer the question. The volume is cubic centimeters.2,142 2,142
ⓑ | |
Step 2. Identify what you are looking for. | the surface area of the solid |
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: Double-check with a calculator. | |
Step 7. Answer the question. | The surface area is 1,034 square centimeters. |
Try It 9.93
Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length
Try It 9.94
Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length
Example 9.48
A rectangular crate has a length of
ⓑ | |
Step 2. Identify what you are looking for. | the surface area of the crate |
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: Check it yourself! | |
Step 7. Answer the question. | The surface area is 3,700 square inches. |
- Answer
-
Step 1 is the same for both ⓐ and ⓑ, so we will show it just once.
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 crate Step 3. Name. Choose a variable to represent it. let = volumeV V Step 4. Translate.
Write the appropriate formula.
Substitute.
V = L W H V = L W H
V = 30 ⋅ 25 ⋅ 20 V = 30 ⋅ 25 ⋅ 20 Step 5. Solve the equation. V = 15, 000 V = 15,000 Step 6. Check: Double check your math. Step 7. Answer the question. The volume is 15,000 cubic inches.
Try It 9.95
A rectangular box has length
Try It 9.96
A rectangular suitcase has length
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
Volume and Surface Area of a Cube
For any cube with sides of length
Example 9.49
A cube is
- Answer
-
Step 1 is the same for both ⓐ and ⓑ, so we will show it just once.
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 cube Step 3. Name. Choose a variable to represent it. let V = volume Step 4. Translate.
Write the appropriate formula.
V = s 3 V = s 3 Step 5. Solve. Substitute and solve. V = ( 2.5 ) 3 V = ( 2.5 ) 3
V = 15.625 V = 15.625 Step 6. Check: Check your work. Step 7. Answer the question. The volume is 15.625 cubic inches.
ⓑ | |
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 9.97
For a cube with side 4.5 meters, find the ⓐ volume and ⓑ surface area of the cube.
Try It 9.98
For a cube with side 7.3 yards, find the ⓐ volume and ⓑ surface area of the cube.
Example 9.50
A notepad cube measures
- Answer
-
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 cube Step 3. Name. Choose a variable to represent it. let V = volume Step 4. Translate.
Write the appropriate formula.
V = s 3 V = s 3 Step 5. Solve the equation. V = 2 3 V = 2 3
V = 8 V = 8 Step 6. Check: Check that you did the calculations
correctly.Step 7. Answer the question. The volume is 8 cubic inches.
ⓑ | |
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 9.99
A packing box is a cube measuring
Try It 9.100
A wall is made up of cube-shaped bricks. Each cube is
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 center 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
Volume and Surface Area of a Sphere
For a sphere with radius
Example 9.51
A sphere has a radius
- Answer
-
Step 1 is the same for both ⓐ and ⓑ, so we will show it just once.
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 sphere Step 3. Name. Choose a variable to represent it. let V = volume Step 4. Translate.
Write the appropriate formula.
V = 4 3 π r 3 V = 4 3 π r 3 Step 5. Solve. V ≈ 4 3 ( 3.14 ) 6 3 V ≈ 4 3 ( 3.14 ) 6 3
V ≈ 904.32 cubic inches V ≈ 904.32 cubic inches Step 6. Check: Double-check your math on a calculator. Step 7. Answer the question. The volume is approximately 904.32 cubic inches.
ⓑ | |
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. |
|
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 9.101
Find the ⓐ volume and ⓑ surface area of a sphere with radius 3 centimeters.
Try It 9.102
Find the ⓐ volume and ⓑ surface area of each sphere with a radius of
Example 9.52
A globe of Earth is in the shape of a sphere with radius
- Answer
-
Step 1. Read the problem. Draw a figure with the
given information and label it.ⓐ 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 )π π
V = 4 3 π r 3 V = 4 3 π r 3
V ≈ 4 3 ( 3.14 ) 14 3 V ≈ 4 3 ( 3.14 ) 14 3 Step 5. Solve. V ≈ 11,488.21 V ≈ 11,488.21 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.
ⓑ | |
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 9.103
A beach ball is in the shape of a sphere with radius of
Try It 9.104
A Roman statue depicts Atlas holding a globe with radius of
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
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,
For the rectangular solid, the area of the base,
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 9.34.
The distance around the edge of the can is the circumference of the cylinder’s base it is also the length
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
Volume and Surface Area of a Cylinder
For a cylinder with radius
Example 9.53
A cylinder has height
- Answer
-
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 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 )π π
V = π r 2 h V = π r 2 h
V ≈ ( 3.14 ) 3 2 ⋅ 5 V ≈ ( 3.14 ) 3 2 ⋅ 5 Step 5. Solve. V ≈ 141.3 V ≈ 141.3 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.
ⓑ | |
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 9.105
Find the ⓐ volume and ⓑ surface area of the cylinder with radius 4 cm and height 7cm.
Try It 9.106
Find the ⓐ volume and ⓑ surface area of the cylinder with given radius 2 ft and height 8 ft.
Example 9.54
Find the ⓐ volume and ⓑ surface area of a can of soda. The radius of the base is
- Answer
-
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 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 )π π
V = π r 2 h V = π r 2 h
V ≈ ( 3.14 ) 4 2 ⋅ 13 V ≈ ( 3.14 ) 4 2 ⋅ 13 Step 5. Solve. V ≈ 653.12 V ≈ 653.12 Step 6. Check: We leave it to you to check. Step 7. Answer the question. The volume is approximately 653.12 cubic centimeters.
ⓑ | |
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 centimeters. |
Try It 9.107
Find the ⓐ volume and ⓑ surface area of a can of paint with radius 8 centimeters and height 19 centimeters. Assume the can is shaped exactly like a cylinder.
Try It 9.108
Find the ⓐ volume and ⓑ surface area of a cylindrical drum with radius 2.7 feet and height 4 feet. Assume the drum is shaped exactly like a cylinder.
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 9.35.
Earlier in this section, we saw that the volume of a cylinder is
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,
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
Example 9.55
Find the volume of a cone with height
- Answer
-
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 )π π
V = 1 3 π r 2 h V = 1 3 π r 2 h
V ≈ 1 3 3.14 ( 2 ) 2 ( 6 ) V ≈ 1 3 3.14 ( 2 ) 2 ( 6 ) Step 5. Solve. V ≈ 25.12 V ≈ 25.12 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 9.109
Find the volume of a cone with height
Try It 9.110
Find the volume of a cone with height
Example 9.56
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
- Answer
-
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 diameter. The radius is 2.5 inches.)π π
V = 1 3 π r 2 h V = 1 3 π r 2 h
V ≈ 1 3 3.14 ( 2.5 ) 2 ( 8 ) V ≈ 1 3 3.14 ( 2.5 ) 2 ( 8 ) Step 5. Solve. V ≈ 52.33 V ≈ 52.33 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 9.111
How many cubic inches of candy will fit in a cone-shaped piñata that is
Try It 9.112
What is the volume of a cone-shaped party hat that is
Summary of Geometry Formulas
The following charts summarize all of the formulas covered in this chapter.
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Section 9.6 Exercises
Practice Makes Perfect
Find Volume and Surface Area of Rectangular Solids
In the following exercises, find ⓐ the volume and ⓑ the surface area of the rectangular solid with the given dimensions.
length
length
length
length
In the following exercises, solve.
Moving van A rectangular moving van has length
Gift box A rectangular gift box has length
Carton A rectangular carton has length
Shipping container A rectangular shipping container has length
In the following exercises, find ⓐ the volume and ⓑ the surface area of the cube with the given side length.
In the following exercises, solve.
Science center Each side of the cube at the Discovery Science Center in Santa Ana is
Museum A cube-shaped museum has sides
Base of statue The base of a statue is a cube with sides
Tissue box A box of tissues is a cube with sides 4.5 inches long. Find its ⓐ volume and ⓑ surface area.
Find the Volume and Surface Area of Spheres
In the following exercises, find ⓐ the volume and ⓑ the surface area of the sphere with the given radius. Round answers to the nearest hundredth.
In the following exercises, solve. Round answers to the nearest hundredth.
Exercise ball An exercise ball has a radius of
Balloon ride The Great Park Balloon is a big orange sphere with a radius of
Golf ball A golf ball has a radius of
Baseball A baseball has a radius of
Find the Volume and Surface Area of a Cylinder
In the following exercises, find ⓐ the volume and ⓑ the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth.
radius
radius
radius
radius
In the following exercises, solve. Round answers to the nearest hundredth.
Coffee can A can of coffee has a radius of
Snack pack A snack pack of cookies is shaped like a cylinder with radius
Barber shop pole A cylindrical barber shop pole has a diameter of
Architecture A cylindrical column has a diameter of
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.
height
height
height
height
In the following exercises, solve. Round answers to the nearest hundredth.
Teepee What is the volume of a cone-shaped teepee tent that is
Popcorn cup What is the volume of a cone-shaped popcorn cup that is
Silo What is the volume of a cone-shaped silo that is
Sand pile What is the volume of a cone-shaped pile of sand that is
Everyday Math
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
- ⓐ find the volume of the large cone.
- ⓑ find the volume of the small cone.
- ⓒ find the volume of the post by subtracting the volume of the small cone from the volume of the large cone.
Ice cream cones A regular ice cream cone is 4 inches tall and has a diameter of
- ⓐ find the volume of the regular ice cream cone.
- ⓑ find the volume of the waffle cone.
- ⓒ how much more ice cream fits in the waffle cone compared to the regular cone?
Writing Exercises
The formulas for the volume of a cylinder and a cone are similar. Explain how you can remember which formula goes with which shape.
Which has a larger volume, a cube of sides of
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?