9.6 Solve Geometry Applications: Volume and Surface Area

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.

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.

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 $\text{length}\times \text{width}\times \text{height}\text{.}$

The volume, $V,$ 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, $B,$ is equal to $\text{length}\times \text{width}\text{.}$

We can substitute $B$ for $L·W$ 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 $4\times 2\times 3$ rectangular solid we started with. See Figure 9.29.

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.

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

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.

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.

Volume and Surface Area of a Rectangular Solid

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

Example 9.47

For a rectangular solid with length $14$ cm, height $17$ cm, and width $9$ cm, find the ⓐ volume and ⓑ surface area.

Solution

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 $V$= volume
Step 4. Translate. Write the appropriate formula. Substitute.	$V=LWH$ $V=\mathrm{14}\cdot 9\cdot 17$
Step 5. Solve the equation.	$V=\mathrm{2,142}$
Step 6. Check We leave it to you to check your calculations.
Step 7. Answer the question.	The volume is $\text{2,142}$ 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 $S$= surface area
Step 4. Translate. Write the appropriate formula. Substitute.	$S=2LH+2LW+2WH$ $S=2(14\cdot 17)+2(14\cdot 9)+2(9\cdot 17)$
Step 5. Solve the equation.	$S=\mathrm{1,034}$
Step 6. Check: Double-check with a calculator.
Step 7. Answer the question.	The surface area is 1,034 square centimeters.

Example 9.48

A rectangular crate has a length of $30$ inches, width of $25$ inches, and height of $20$ inches. Find its ⓐ volume and ⓑ surface area.

Solution

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 $V$= volume
Step 4. Translate. Write the appropriate formula. Substitute.	$V=LWH$ $V=30\cdot 25\cdot 20$
Step 5. Solve the equation.	$V=\mathrm{15,000}$
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 $S$= surface area
Step 4. Translate. Write the appropriate formula. Substitute.	$S=2LH+2LW+2WH$ $S=2(30\cdot 20)+2(30\cdot 25)+2(25\cdot 20)$
Step 5. Solve the equation.	$S=\mathrm{3,700}$
Step 6. Check: Check it yourself!
Step 7. Answer the question.	The surface area is 3,700 square inches.

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:

$$\begin{array}{ccccc}V=LWH\hfill & & & & S=2LH+2LW+2WH\hfill \\ V=s·s·s\hfill & & & & S=2s·s+2s·s+2s·s\hfill \\ V=s^3\hfill & & & & S=2s^2+2s^2+2s^2\hfill \\ & & & & S=6s^2\hfill \end{array}$$

So for a cube, the formulas for volume and surface area are $V=s^3$ and $S=6s^2.$

Volume and Surface Area of a Cube

For any cube with sides of length $s,$

Example 9.49

A cube is $2.5$ inches on each side. Find its ⓐ volume and ⓑ surface area.

Solution

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.	$V=s^3$
Step 5. Solve. Substitute and solve.	$V={(2.5)}^3$ $V=15.625$
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.	$S=6s^2$
Step 5. Solve. Substitute and solve.	$S=6\cdot {(2.5)}^2$ $S=37.5$
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 $2$ inches on each side. Find its ⓐ volume and ⓑ surface area.

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 cube
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.	$V=2^3$ $V=8$
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.	$S=6s^2$
Step 5. Solve the equation.	$S=6\cdot 2^2$ $S=24$
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 $4$ feet on each side. Find its ⓐ volume and ⓑ surface area.

Try It 9.100

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

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 $\pi$ with $3.14.$

Volume and Surface Area of a Sphere

For a sphere with radius $r\text{:}$

Example 9.51

A sphere has a radius $6$ inches. Find its ⓐ volume and ⓑ surface area.

Solution

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.	$V=\frac{4}{3}\pi r^3$
Step 5. Solve.	$V\approx \frac{4}{3}(3.14)6^3$ $V\approx 904.32\text{cubic inches}$
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 cube
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula.	$S=4\pi r^2$
Step 5. Solve.	$S\approx 4(3.14)6^2$ $S\approx 452.16$
Step 6. Check: Double-check your math on a calculator
Step 7. Answer the question.	The surface area is approximately 452.16 square inches.

Example 9.52

A globe of Earth is in the shape of a sphere with radius $14$ centimeters. Find its ⓐ volume and ⓑ 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.

ⓐ
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 $\pi$)	$V=\frac{4}{3}\pi r^3$ $V\approx \frac{4}{3}(3.14){14}^3$
Step 5. Solve.	$V\approx \mathrm{11,488.21}$
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 $\pi$)	$S=4\pi r^2$ $S\approx 4(3.14){14}^2$
Step 5. Solve.	$S\approx 2461.76$
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.

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 $h$ of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, $h$ , will be perpendicular to the bases.

Figure 9.32 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, $V=Bh$ , can also be used to find the volume of a cylinder.

For the rectangular solid, the area of the base, $B$ , is the area of the rectangular base, length × width. For a cylinder, the area of the base, $B,$ is the area of its circular base, $\pi r^2.$ Figure 9.33 compares how the formula $V=Bh$ is used for rectangular solids and cylinders.

Figure 9.33 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 9.34.

Figure 9.34 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 $L$ of the rectangular label. The height of the cylinder is the width $W$ 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 $r$ and height $h,$ is

Volume and Surface Area of a Cylinder

For a cylinder with radius $r$ and height $h:$

Example 9.53

A cylinder has height $5$ centimeters and radius $3$ centimeters. Find the ⓐ volume and ⓑ surface area.

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 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 $\pi$)	$V=\pi r^{2}h$ $V\approx (3.14)3^2\cdot 5$
Step 5. Solve.	$V\approx 141.3$
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 $\pi$)	$S=2\pi r^2+2\pi rh$ $S\approx 2(3.14)3^2+2(3.14)(3)5$
Step 5. Solve.	$S\approx 150.72$
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.

Example 9.54

Find the ⓐ volume and ⓑ surface area of a can of soda. The radius of the base is $4$ centimeters and the height is $13$ centimeters. 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.

ⓐ
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 $\pi$)	$V=\pi r^{2}h$ $V\approx (3.14)4^2\cdot 13$
Step 5. Solve.	$V\approx 653.12$
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 $\pi$)	$S=2\pi r^2+2\pi rh$ $S\approx 2(3.14)4^2+2(3.14)(4)13$
Step 5. Solve.	$S\approx 427.04$
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.

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.

Figure 9.35 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 $V=\text{π}{r}^{2}h.$ We can think of a cone as part of a cylinder. Figure 9.36 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.

Figure 9.36 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, $\text{π}{r}^2$ , for $B$ 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 $r$ and height $h$.

Example 9.55

Find the volume of a cone with height $6$ inches and radius of its base $2$ 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 $\pi$)	$V=\frac{1}{3}\pi r^{2}h$ $V\approx \frac{1}{3}3.14{(2)}^2(6)$
Step 5. Solve.	$V\approx 25.12$
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.

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 $8$ inches tall and $5$ inches in diameter? 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 $\pi$, and notice that we were given the distance across the circle, which is its diameter. The radius is 2.5 inches.)	$V=\frac{1}{3}\pi r^{2}h$ $V\approx \frac{1}{3}3.14{(2.5)}^2(8)$
Step 5. Solve.	$V\approx 52.33$
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.

Summary of Geometry Formulas

The following charts summarize all of the formulas covered in this chapter.