9.6 Solve Geometry Applications: Volume and Surface Area

Learning objectives

Learning objectives

• 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

1. 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.

2. Step 2. Identify what you are looking for.

3. Step 3. Name what you are looking for. Choose a variable to represent that quantity.

4. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.

5. Step 5. Solve the equation using good algebra techniques.

6. Step 6. Check the answer in the problem and make sure it makes sense.

7. 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 $4$ units, width $2$ units, and height $3$ units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Altogether there are $24$ cubic units. Notice that $24$ is the $\mathrm{length}\,\times\,\mathrm{width}\,\times\,\mathrm{height}.$

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 $\mathrm{length}\times\mathrm{width}.$

We can substitute $B$ for $L\cdot{}W$ 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.

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 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.

Example 9.47

Solution

1. State the volume formula. A rectangular solid has volume $V=LWH$. Substituting the dimensions gives $V=(14)(9)(17)=126(17)=2142$ cubic centimeters.
2. State the surface-area formula. Use $S=2LH+2LW+2WH$, so $S=2(14)(17)+2(14)(9)+2(9)(17)$.
3. Evaluate the surface area. The three doubled face areas are $476$, $252$, and $306$ square centimeters, which sum to $1034$ square centimeters.
4. Check the units. Volume describes the enclosed space and has cubic units, while surface area describes the exterior faces and has square units.
5. State both answers. The volume is $2142$ cubic centimeters, and the surface area is $1034$ square centimeters.

Takeaway: Use $V=LWH$ for a rectangular solid's volume and $S=2LH+2LW+2WH$ for its surface area; the dimensions $14$ by $9$ by $17$ give $2142$ cubic centimeters and $1034$ square centimeters.

Example 9.48

Solution

1. Calculate the volume. Use $V=LWH$ with the crate dimensions to obtain $V=(30)(25)(20)=750(20)=15000$ cubic inches.
2. Set up the surface area. The formula $S=2LH+2LW+2WH$ becomes $S=2(30)(20)+2(30)(25)+2(25)(20)$.
3. Evaluate the three face pairs. The contributions are $1200$, $1500$, and $1000$ square inches, giving $S=1200+1500+1000=3700$ square inches.
4. Check the units. The enclosed volume is measured in cubic inches, while the six exterior faces are measured in square inches.
5. State both answers. The crate has volume $15000$ cubic inches and surface area $3700$ square inches.

Takeaway: A crate with dimensions $30$ by $25$ by $20$ has volume $15000$ cubic inches and surface area $3700$ 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:

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

Example 9.49

Solution

1. Calculate the cube volume. A cube has side length $s$, so $V=s^3$. Substituting $s=2.5$ gives $V=(2.5)^3=15.625$ cubic inches.
2. Calculate the cube surface area. The formula $S=6s^2$ gives $S=6(2.5)^2=6(6.25)=37.5$ square inches.
3. Check the formulas. The volume is the side length multiplied by itself three times, while the surface area counts six congruent square faces.
4. State both answers. The cube has volume $15.625$ cubic inches and surface area $37.5$ square inches.

Takeaway: For a cube with side $s$, use $V=s^3$ and $S=6s^2$; a side of $2.5$ inches gives volume $15.625$ and surface area $37.5$ square inches.

Example 9.50

Solution

1. Calculate the cube volume. With side length $s=2$ inches, use $V=s^3$ to obtain $V=2^3=8$ cubic inches.
2. Calculate the cube surface area. Use $S=6s^2$ to obtain $S=6(2^2)=6(4)=24$ square inches.
3. Check the formulas. The three factors in $s^3$ describe the cube's length, width, and height, while $6s^2$ counts its six square faces.
4. State both answers. The notepad cube has volume $8$ cubic inches and surface area $24$ square inches.

Takeaway: A cube's volume is $s^3$ and its surface area is $6s^2$, so a side of $2$ inches gives volume $8$ cubic inches and surface area $24$ square inches.

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 $3.14.$

Example 9.51

Solution

1. Calculate the sphere volume. Use $V=\frac{4}{3}\pi r^3$ with $r=6$ to get $V\approx\frac{4}{3}(3.14)(6^3)=904.32$ cubic inches.
2. Calculate the sphere surface area. Use $S=4\pi r^2$ to get $S\approx4(3.14)(6^2)=452.16$ square inches.
3. Check the powers and units. The volume uses $r^3$ and cubic inches, whereas the surface area uses $r^2$ and square inches.
4. State both answers. The sphere has volume approximately $904.32$ cubic inches and surface area approximately $452.16$ square inches.

Takeaway: For a sphere of radius $r$, use $V=\frac{4}{3}\pi r^3$ and $S=4\pi r^2$; radius $6$ inches gives approximately $904.32$ cubic inches and $452.16$ square inches.

Example 9.52

Solution

1. Calculate the sphere volume. Substitute $r=14$ into $V=\frac{4}{3}\pi r^3$ to get $V\approx\frac{4}{3}(3.14)(14^3)=11488.21$ cubic inches after rounding.
2. Calculate the sphere surface area. Substitute $r=14$ into $S=4\pi r^2$ to get $S\approx4(3.14)(14^2)=2461.76$ square inches after rounding.
3. Check the powers and units. The volume formula cubes the radius and uses cubic inches, while the surface-area formula squares it and uses square inches.
4. State both answers. The globe has volume approximately $11488.21$ cubic inches and surface area approximately $2461.76$ square inches.

Takeaway: A sphere of radius $14$ inches has volume approximately $11488.21$ cubic inches and surface area 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.

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.

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 $L$ 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

Example 9.53

Solution

1. Calculate the cylinder volume. Use $V=\pi r^2h$ with $r=3$ and $h=5$ to get $V\approx(3.14)(3^2)(5)=141.3$ cubic inches.
2. Calculate the cylinder surface area. Use $S=2\pi r^2+2\pi rh$ to get $S\approx2(3.14)(3^2)+2(3.14)(3)(5)=150.72$ square inches.
3. Check the two bases and curved surface. The first term in the surface-area formula counts both circular bases, and the second counts the curved lateral surface.
4. State both answers. The cylinder has volume approximately $141.3$ cubic inches and surface area approximately $150.72$ square inches.

Takeaway: For a cylinder, use $V=\pi r^2h$ and $S=2\pi r^2+2\pi rh$; radius $3$ and height $5$ inches give approximately $141.3$ cubic inches and $150.72$ square inches.

Example 9.54

Solution

1. Calculate the can volume. Use $V=\pi r^2h$ with $r=4$ and $h=13$ to get $V\approx(3.14)(4^2)(13)=653.12$ cubic centimeters.
2. Calculate the can surface area. Use $S=2\pi r^2+2\pi rh$ to get $S\approx2(3.14)(4^2)+2(3.14)(4)(13)=427.04$ square centimeters.
3. Check the cylinder structure. The volume uses the circular base area times the height, while the surface area includes two bases and the curved side.
4. State both answers. The can has volume approximately $653.12$ cubic centimeters and surface area approximately $427.04$ square centimeters.

Takeaway: A cylindrical can with radius $4$ and height $13$ centimeters has volume approximately $653.12$ cubic centimeters and surface area 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.

Earlier in this section, we saw that the volume of a cylinder is $V=\pi{}{r}^{2}h.$ 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.

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, $\pi{}{r}^{2}$ , for $B$ to get the formula for volume of a cone.

In this course, we will only find the volume of a cone, and not its surface area.

Example 9.55

Solution

1. State the cone volume formula. Use $V = \frac{1}{3}\pi r^2h$, where $r$ is the base radius and $h$ is the perpendicular height.
2. Substitute the dimensions. With $r = 2$ inches and $h = 6$ inches, the formula becomes $V = \frac{1}{3}\pi(2)^2(6)$.
3. Simplify the product. Since $(2)^2(6) = 24$, we obtain $V = 8\pi$. Using $\pi \approx 3.14$ gives $V \approx 25.12$ cubic inches.
4. Confirm the diagram. The figure shows a cone with base radius $2$ inches and height $6$ inches, matching the values used in the formula. Therefore the cone's volume is approximately $25.12$ cubic inches.

Takeaway: The volume of a cone equals one third of $\pi r^2h$ because it has one third the volume of the matching cylinder.

Example 9.56

Solution

1. Find the radius from the diameter. The circular opening is $5$ inches across, so $r = \frac{5}{2} = 2.5$ inches; the cone's height is $h = 8$ inches.
2. State and substitute the cone formula. Use $V = \frac{1}{3}\pi r^2h$ to write $V = \frac{1}{3}(3.14)(2.5)^2(8)$.
3. Evaluate the expression. The squared radius is $6.25$, and $\frac{1}{3}(3.14)(6.25)(8) \approx 52.33$.
4. Interpret the diagram. The illustration depicts an inverted cone with height $8$ inches and a circular opening measuring $5$ inches, the quantity identified as the diameter in the problem. Thus the conic wrap's volume is approximately $52.33$ cubic inches.

Takeaway: Convert a cone's diameter to radius before substituting, because the volume formula contains $r^2$.

Summary of Geometry Formulas

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

Section 9.6 Exercises

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.

In the following exercises, solve.

In the following exercises, find ⓐ the volume and ⓑ the surface area of the cube with the given side length.

In the following exercises, solve.

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.

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.

In the following exercises, solve. Round answers to the nearest hundredth.

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.

In the following exercises, solve. Round answers to the nearest hundredth.

Everyday Math

Writing Exercises

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?

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

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

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

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

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

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

1. 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.

2. Find the ⓐ volume and ⓑ surface area of a sphere with radius 3 centimeters.

3. Find the ⓐ volume and ⓑ surface area of each sphere with a radius of $1$ foot

1. A beach ball is in the shape of a sphere with radius of $9$ inches. Find its ⓐ volume and ⓑ surface area.

2. A Roman statue depicts Atlas holding a globe with radius of $1.5$ feet. Find the ⓐ volume and ⓑ surface area of the globe.

3. Find the ⓐ volume and ⓑ surface area of the cylinder with radius 4 cm and height 7cm.

1. Find the ⓐ volume and ⓑ surface area of the cylinder with given radius 2 ft and height 8 ft.

2. 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.

3. 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.

1. Find the volume of a cone with height $7$ inches and radius $3$ inches

2. Find the volume of a cone with height $9$ centimeters and radius $5$ centimeters

3. How many cubic inches of candy will fit in a cone-shaped piñata that is $18$ inches long and $12$ inches across its base? Round the answer to the nearest hundredth.

1. What is the volume of a cone-shaped party hat that is $10$ inches tall and $7$ inches across at the base? Round the answer to the nearest hundredth.

2. length $2$ meters, width $1.5$ meters, height $3$ meters

3. length $3.5$ yards, width $2.1$ yards, height $2.4$ yards

1. Moving van A rectangular moving van has length $16$ feet, width $8$ feet, and height $8$ feet. Find its ⓐ volume and ⓑ surface area.

2. Carton A rectangular carton has length $21.3$ cm, width $24.2$ cm, and height $6.5$ cm. Find its ⓐ volume and ⓑ surface area.

3. $5$ centimeters

1. $10.4$ feet

2. Science center Each side of the cube at the Discovery Science Center in Santa Ana is $64$ feet long. Find its ⓐ volume and ⓑ surface area.

3. Base of statue The base of a statue is a cube with sides $2.8$ meters long. Find its ⓐ volume and ⓑ surface area.

1. $3$ centimeters

2. $7.5$ feet

3. Exercise ball An exercise ball has a radius of $15$ inches. Find its ⓐ volume and ⓑ surface area.

1. Golf ball A golf ball has a radius of $4.5$ centimeters. Find its ⓐ volume and ⓑ surface area.

2. radius $3$ feet, height $9$ feet

3. radius $1.5$ meters, height $4.2$ meters

1. Coffee can A can of coffee has a radius of $5$ cm and a height of $13$ cm. Find its ⓐ volume and ⓑ surface area.

2. Barber shop pole A cylindrical barber shop pole has a diameter of $6$ inches and height of $24$ inches. Find its ⓐ volume and ⓑ surface area.

3. height $9$ feet and radius $2$ feet

1. height $12.4$ centimeters and radius $5$ cm

2. Teepee What is the volume of a cone-shaped teepee tent that is $10$ feet tall and $10$ feet across at the base?

3. Silo What is the volume of a cone-shaped silo that is $50$ feet tall and $70$ feet across at the base?

1. 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 $30$ feet tall with base radius $1$ foot. The smaller cone is $10$ feet tall with base radius of $0.5$ feet. To the nearest tenth,

1. ⓐ find the volume of the large cone.

2. ⓑ find the volume of the small cone.

3. ⓒ find the volume of the post by subtracting the volume of the small cone from the volume of the large cone.

2. The formulas for the volume of a cylinder and a cone are similar. Explain how you can remember which formula goes with which shape.