3.5: Solve Geometry Applications- Volume and Surface Area
- Page ID
- 152039
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- 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
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.
Problem Solving Strategy for Geometry Applications
- Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
- Identify what you are looking for.
- Name what you are looking for. Choose a variable to represent that quantity.
- Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- 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 3.5.1). 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.
Each crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in Figure 3.5.2 has length units, width units, and height units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.
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 shape from Figure 3.5.2.
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.
S = (front + back) + (left side + right side) + (top + bottom)
S = (2 · front) + (2 · left side) + (2 · top)
S = 2 · 12 + 2 · 6 + 2 · 8
S = 24 + 12 + 16
S = 52 square units
The surface area of Figure 3.5.2 is 52 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 two dimensions, either length and width, length and height, or width and height (see Figure 3.5.3). 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 width and height
Example 3.5.1
For a rectangular solid with length cm, height cm, and width cm, find the ⓐ volume and ⓑ surface area.
- 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 = 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 volume is 2,142 cubic centimeters. ⓑ 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 centimeters.
Your Turn 3.5.1
Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length feet, width feet, and height feet.
Example 3.5.2
A rectangular crate has a length of inches, width of inches, and height of inches. Find its ⓐ volume and ⓑ surface area.
- 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 = 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. ⓑ 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. Double check your math. Step 7. Answer the question. The surface area is 3,700 square inches.
Your Turn 3.5.2
A rectangular box has length feet, width feet, and height feet. Find its ⓐ volume and ⓑ surface area.
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:
Volume and Surface Area of a Cube
For any cube with sides of length
Example 3.5.3
A cube is inches on each side. Find its ⓐ volume and ⓑ surface area.
- 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.
Step 5. Solve. Substitute and solve.
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.
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.
Your Turn 3.5.3
For a cube with side 4.5 meters, find the ⓐ volume and ⓑ surface area of the cube.
Example 3.5.4
A notepad cube measures inches on each side. Find its ⓐ volume and ⓑ surface area.
- 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.
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. ⓑ 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.
Your Turn 3.5.4
A packing box is a cube measuring feet on each side. Find its ⓐ volume and ⓑ surface area.
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 again use the button on our calculator to get accurate answers.
Volume and Surface Area of a Sphere
For a sphere with radius
Example 3.5.5
A sphere has a radius of inches. Find its ⓐ volume and ⓑ surface area. Round your answers to the nearest hundredth.
- 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.
Step 5. Solve.
Step 6. Check. Double-check your math on a calculator. Step 7. Answer the question. The volume is approximately 904.78 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. We leave it to you to check your calculations. Step 7. Answer the question. The surface area is approximately 452.39 square inches.
Your Turn 3.5.5
Find the ⓐ volume and ⓑ surface area of a sphere with radius 3 centimeters. Round your answers to the nearest tenth.
Example 3.5.6
A globe of Earth is in the shape of a sphere with radius inches. Find its ⓐ volume and ⓑ surface area. Round the answers to the nearest hundredth.
- 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.
Step 5. Solve. Step 6. Check: We leave it to you to check your calculations. Step 7. Answer the question. The volume is approximately 11,494.03 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.
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 2463.01 square inches.
Your Turn 3.5.6
A beach ball is in the shape of a sphere with radius of inches. Find its ⓐ volume and ⓑ surface area. Round your answers to the nearest hundredth.
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.
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.
Figure 3.5.5 compares how the formula is used for rectangular solids and cylinders. In a rectangular solid, the base is a rectangle. In a cylinder, the base is a circle.
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 3.5.6.
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 3.5.7
A cylinder has height inches and radius inches. Find the ⓐ volume and ⓑ surface area. Round your answers to the nearest tenth.
- 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.
Step 5. Solve. Step 6. Check. We leave it to you to check the calculations. Step 7. Answer the question. The volume is approximately 141.4 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.
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 150.8 square inches.
Your Turn 3.5.7
Find the ⓐ volume and ⓑ surface area of the cylinder with radius 4 cm and height 7 cm. Round your answers to the nearest hundredth.
Example 3.5.8
Find the ⓐ volume and ⓑ surface area of a can of soda. The radius of the base is centimeters and the height is centimeters. Assume the can is shaped exactly like a cylinder. Round your answers to the nearest hundredth.
- 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.
Step 5. Solve. Step 6. Check: We leave it to you to check. Step 7. Answer the question. The volume is approximately 653.45 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.
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 427.26 square centimeters.
Your Turn 3.5.8
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.
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 3.5.7.
Earlier in this section, we saw that the volume of a cylinder is . Figure 3.5.8 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.
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 3.5.9
Find the volume of a cone with height inches and radius of its base inches. Round your answer to the nearest thousandth.
- 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.
Step 5. Solve. Step 6. Check. We leave it to you to check your calculations. Step 7. Answer the question. The volume is approximately 25.133 cubic inches.
Your Turn 3.5.9
Find the volume of a cone with height inches and radius inches. Round your answer to the nearest tenth.
Example 3.5.10
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 radius? Round the answer to the nearest hundredth.
- 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.
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 209.44 cubic inches.
Your Turn 3.5.10
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.
Summary of Geometry Formulas
The following charts summarize all of the formulas covered in this chapter.