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Mathematics LibreTexts

9.6: Solve Geometry Applications: Volume and Surface Area (Part 1)

Skills to Develop

  • 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!

Before you get started, take this readiness quiz.

  1. Evaluate x3 when x = 5. If you missed this problem, review Example 2.15.
  2. Evaluate 2x when x = 5. If you missed this problem, review Example 2.16.
  3. Find the area of a circle with radius \(\frac{7}{2}\). 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.

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.

CNX_BMath_Figure_09_06_001.jpg

Figure 9.28 - 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 9.29 has length 4 units, width 2 units, and height 3 units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

CNX_BMath_Figure_09_06_002.jpg

Figure 9.29 - Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This 4 by 2 by 3 rectangular solid has 24 cubic units.

Altogether there are 24 cubic units. Notice that 24 is the length × width × height.

CNX_BMath_Figure_09_06_002_img.jpg

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

$$V= LWH$$

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, B, is equal to length × width.

$$B = L \cdot W$$

We can substitute B for L • W in the volume formula to get another form of the volume formula.

$$\begin{split} V &= \textcolor{red}{L \cdot W} \cdot H \\ V &= \textcolor{red}{(L \cdot W)} \cdot H \\ V &= \textcolor{red}{B} h \end{split}$$

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the 4 × 2 × 3 rectangular solid we started with. See Figure 9.29.

CNX_BMath_Figure_09_06_004_img.jpg

Figure 9.30

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.

$$\begin{split} A_{front} &= L \times W \qquad A_{side} = L \times W \qquad A_{top} = L \times W \\ A_{front} &= 4 \cdot 3 \qquad \quad \; A_{side} = 2 \cdot 3 \qquad \quad \; A_{top} = 4 \cdot 2 \\ A_{front} &= 12 \qquad \qquad A_{side} = 6 \qquad \qquad \; \; A_{top} = 8 \end{split}$$

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

$$\begin{split} S &= (front + back)+(left\; side + right\; side) + (top + bottom) \\ S &= (2 \cdot front) + (2 \cdot left\; side) + (2 \cdot top) \\ S &= 2 \cdot 12 + 2 \cdot 6 + 2 \cdot 8 \\ S &= 24 + 12 + 16 \\ S &= 52\; sq.\; units \end{split}$$

The surface area S of the rectangular solid shown in Figure 9.30 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 length and its width (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.

$$S = 2LH + 2LW + 2WH$$

CNX_BMath_Figure_09_06_005.jpg

Figure 9.31 - For each face of the rectangular solid facing you, there is another face on the opposite side. There are 6 faces in all.

Definition: Volume and Surface Area of a Rectangular Solid

For a rectangular solid with length L, width W, and height H:

CNX_BMath_Figure_09_06_006_img.jpg

Example 9.47:

For a rectangular solid with length 14 cm, height 17 cm, and width 9 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. CNX_BMath_Figure_09_06_038_img-01.png

(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 V = volume
Step 4. Translate. Write the appropriate formula. Substitute. $$\begin{split} V &= LWH \\ V &= 14 \cdot 9 \cdot 9 \cdot 17 \end{split}$$
Step 5. Solve the equation. $$V = 2,142$$
Step 6. Check.  We leave it to you to check your calculations.
Step 7. Answer the question. The surface area is 1,034 square centimeters.

(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 S = surface area
Step 4. Translate. Write the appropriate formula. Substitute. $$\begin{split} S &= 2LH + 2LW + 2WH \\ S &= 2(14 \cdot 17) + 2(14 \cdot 9) + 2(9 \cdot 17) \end{split}$$
Step 5. Solve the equation. $$S = 1,034$$
Step 6. Check.  Double-check with a calculator.
Step 7. Answer the question. The surface area is 1,034 square centimeters.

Exercise 9.93:

Find the (a) volume and (b) surface area of rectangular solid with the: length 8 feet, width 9 feet, and height 11 feet.

Exercise 9.94:

Find the (a) volume and (b) surface area of rectangular solid with the: length 15 feet, width 12 feet, and height 8 feet.

Example 9.48:

A rectangular crate has a length of 30 inches, width of 25 inches, and height of 20 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. CNX_BMath_Figure_09_06_039_img-01.png

(a)

Step 2. Identify what you are looking for. the volume of the crate
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. $$\begin{split} V &= LWH \\ V &= 30 \cdot 25 \cdot 20 \end{split}$$
Step 5. Solve the equation. $$V = 15,000$$
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. the surface area of the crate
Step 4. Translate. Write the appropriate formula. Substitute. $$\begin{split} S &= 2LH + 2LW + 2WH \\ S &= 2(30 \cdot 20) + 2(30 \cdot 25) + 2(25 \cdot 20) \end{split}$$
Step 5. Solve the equation. $$S = 3,700$$
Step 6. Check.  Check it yourself!
Step 7. Answer the question. The surface area is 3,700 square inches.

Exercise 9.95:

A rectangular box has length 9 feet, width 4 feet, and height 6 feet. Find its (a) volume and (b) surface area.

Exercise 9.96:

A rectangular suitcase has length 22 inches, width 14 inches, and height 9 inches. Find its (a) volume and (b) 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:

$$\begin{split} V &= LWH \quad \; S = 2LH + 2LW + 2WH \\ V &= s \cdot s \cdot s \quad S = 2s \cdot s + 2s \cdot s + 2s \cdot s \\ V &= s^{3} \qquad \quad S = 2s^{2} + 2s^{2} + 2s^{2} \\ &\qquad \qquad \quad \; S = 6s^{2} \end{split}$$

So for a cube, the formulas for volume and surface area are V = s3 and S = 6s2.

Definition: Volume and Surface Area of a Cube

For any cube with sides of length s,

CNX_BMath_Figure_09_06_010_img.jpg

Example 9.49:

A cube is 2.5 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. CNX_BMath_Figure_09_06_040_img-01.png

(a)

Step 2. Identify what you are looking for. the volume of the crate
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate. Write the appropriate formula. $$V = s^{3}$$
Step 5. Solve. Substitute and solve. $$\begin{split} V &= (2.5)^{3} \\ V &= 15.625 \end{split}$$
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 crate
Step 3. Name. Choose a variable to represent it. the surface area of the crate
Step 4. Translate. Write the appropriate formula. $$S = 6s^{2}$$
Step 5. Solve. Substitute and solve. $$\begin{split} S &= 6 \cdot (2.5)^{2} \\ S &= 37.5 \end{split}$$
Step 6. Check.  The check is left to you.
Step 7. Answer the question. The surface area is 37.5 square inches.

Exercise 9.97:

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

Exercise 9.98:

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

Example 9.50:

A notepad cube measures 2 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. CNX_BMath_Figure_09_06_041_img-01.png

(a)

Step 2. Identify what you are looking for. the volume of the crate
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate. Write the appropriate formula. $$V = s^{3}$$
Step 5. Solve the equation. $$\begin{split} V &= 2^{3} \\ V &= 8 \end{split}$$
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 crate
Step 3. Name. Choose a variable to represent it. the surface area of the crate
Step 4. Translate. Write the appropriate formula. $$S = 6s^{2}$$
Step 5. Solve the equation. $$\begin{split} S &= 6 \cdot 2^{2} \\ S &= 24 \end{split}$$
Step 6. Check.  The check is left to you.
Step 7. Answer the question. The surface area is 24 square inches.

Exercise 9.99:

A packing box is a cube measuring 4 feet on each side. Find its (a) volume and (b) surface area.

Exercise 9.100:

A wall is made up of cube-shaped bricks. Each cube is 16 inches on each side. Find the (a) volume and (b) surface area of each cube.

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