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1.4 Solve Geometry Applications: Volume and Surface Area

Learning Objectives

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

How To

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 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.

This is an image of a wooden crate.
Figure 1 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 2 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.
A rectangular with colored layers composed of 8 cubes, measuring 2 by 4. Each layer has 8 cubic units.
Figure 2

Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This 4 by 3 by 2 rectangular solid has 24 cubic units.  Altogether there are 24 cubic units. Notice that 24 is the length × width × height.

Substitute into the formula V=LWH and get 24=4*2*3.The volume ([latex]V[/latex]) of any rectangular solid is the product of the length ([latex]L[/latex]), width ([latex]W[/latex]), and height ([latex]H[/latex]).

[latex]V=LWH[/latex]

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 ([latex]B[/latex]) is equal to the length multiplied by the width.

[latex]B=LW[/latex]

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

[latex]\begin{array}{rl}\hfill V&=\textcolor{red}{L\cdot W}\cdot H \hfill \\ \hfill V &= \textcolor{red}{B}H \hfill \end{array}[/latex]

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the 4 by 3 by 2 rectangular that we started with.

Rectangular solid with 3 layers of 2×4 cubes; steps show V=Bh=base(4×2)×3=8×3=24 cubic units.
Figure 3
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.

[latex]\begin{array}{rlrlrl}\hfill A_{front} &= L\times W\hfill & \hfill A_{side} &= H\times W\hfill & \hfill A_{top} &= H\times L \hfill \\ \hfill A_{front} &= 4\cdot 3 \hfill & \hfill A_{side} &= 2\cdot 3 \hfill & \hfill A_{top} &= 4\cdot 2 \hfill \\ \hfill A_{front} &= 12\hfill & \hfill A_{side} &= 6\hfill & \hfill A_{top}&=8\hfill \end{array}[/latex]

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

[latex]\begin{gather*} S = (\text{front+back})+(\text{left side + right side})+(\text{top + bottom})\\  S=(2\cdot\text{front})+(2\cdot \text{left side})+(2\cdot \text{top}) \\ S=2\cdot 12 + 2\cdot 6 + 2\cdot 8 \\  S=24+12+16 \\ S=52\text{ sq. units} \end{gather*}[/latex]

The surface area, [latex]S[/latex], of the rectangular solid shown in Figure 3 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 4).

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.

A rectangular solid is shown. The sides are labeled L, W, and H. One face is labeled LW and another is labeled WH.
Figure 4

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:

A rectangular with sides L, W, and H. Beside it is Volume: V equals LWH equals BH. Below that is Surface Area: S equals 2LH plus 2LW plus 2WH.
Figure 5

Finding Volume and Surface Area of Rectangular Solid

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

Show Solution to (a)
Explanation Steps
Step 1. Read the problem. Draw the figure and
 label it with the given information.		 A rectangular prism with 14 along the bottom front edge, 9 along the bottom right edge, and 17 along the back right edge.
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.		 [latex]\begin{array}{rl}\hfill V &=LWH\hfill \\ \hfill V &= 14\cdot 9\cdot 17 \hfill\end{array}[/latex]
Step 5. Solve the equation. [latex]V=2{,}142[/latex]
Step 6. Check
We leave it to you to check your calculations.
Step 7. Answer the question. The volume is 2,142 cubic centimeters.
Show Solution to (b)
Explanation Steps
Step 1. Use the same figure from the previous step. A rectangular prism with 14 along the bottom front edge, 9 along the bottom right edge, and 17 along the back right edge.
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.		 [latex]\begin{array}{rl}\hfill S &= 2LH+2LW+2WH\hfill \\ \hfill S &= 2(14\cdot 17)+2(14\cdot 9)+2(9\cdot 17)\hfill \end{array}[/latex]
Step 5. Solve the equation. [latex]S=1{,}034[/latex]
Step 6. Check: Double-check with a calculator.
Step 7. Answer the question. The surface area is 1,034 square centimeters.

Try It

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

Show Solution

(a) 792 cubic feet; (b) 518 square feet

Try It

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

Show Solution

(a) 1440 cubic feet; (b) 792 square feet

Finding Volume and Surface Area of Rectangular Solid

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.

Show Solution to (a)
Explanation Steps
Step 1. Read the problem. Draw the figure and
label it with the given information.		 A rectangular prism with 30 along the bottom front edge, 25 along the bottom right edge, and 20 along the back right edge.
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.		 [latex]\begin{array}{rl}\hfill V &=LWH \hfill \\ \hfill V &= 30\cdot 25\cdot 20\hfill \end{array}[/latex]
Step 5. Solve the equation. [latex]V=15{,}000[/latex]
Step 6. Check: Double check your math.
Step 7. Answer the question. The volume is 15,000 cubic inches.
Show Solution

Try It

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

Show Solution

(a) 216 cubic feet; (b) 228 square feet

Try It

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

Show Solution

(a) 2,772 cubic inches; (b) 1,264 square inches

Volume and Surface Area of a Cube

A cube is a rectangular solid whose length, width, and height are equal.  Substituting [latex]s[/latex] for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

[latex]\begin{array}{rlcrl} \hfill V&=LWH\hfill & \hfill \qquad\qquad\hfill & \hfill S &=2LH+2LW+2WH\hfill \\ \hfill V &= s\cdot s\cdot s\hfill & \hfill & \hfill S &= 2s\cdot s+2s\cdot s+2s\cdot s\hfill \\ \hfill V &= s^3\hfill &\hfill &\hfill S&=6s^2\hfill \end{array}[/latex]

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

Volume and Surface Area of a Cube

For any cube with sides of length [latex]s[/latex]:

A cube with sides s. Volume: V equals s cubed. Surface Area: S equals 6 times s squared.
Figure 6

Finding Volume and Surface Area of a Cube

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

Show Solution to (a)
Explanation Steps
Step 1. Read the problem. Draw the figure and
label it with the given information.		 A cube with length, width, and height of 2.5.
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.		 [latex]V=s^3[/latex]
Step 5. Solve. Substitute and solve. [latex]\begin{array}{rl} \hfill V &=(2.5)^3\hfill \\ \hfill V &=15.625\hfill \end{array}[/latex]
Step 6. Check: Check your work.
Step 7. Answer the question. The volume is 15.625 cubic inches.
Show Solution
Explanation Steps
Step 1. Use the figure from the previous step. A cube with length, width, and height of 2.5.
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.		 [latex]S=6s^2[/latex]
Step 5. Solve. Substitute and solve. [latex]\begin{array}{rl}\hfill S&=6\cdot (2.5)^2\hfill \\ \hfill S &=37.5\hfill \end{array}[/latex]
Step 6. Check: The check is left to you.
Step 7. Answer the question. The surface area is 37.5 square inches.

Try It

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

Show Solution

(a) 91.125 cubic meters; (b) 121.5 square meters

Try It

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

Show Solution

(a) 389.017 cubic yards; (b) 319.74 square yards

Finding Volume and Surface Area of a Cube

A notepad cube measures 2 inches on each side. Find its (a) volume and (b) surface area.

Show Solution to (a)
Explanation Steps
Step 1. Read the problem. Draw the figure and
label it with the given information.		 A cube with length, width, and height labeled as 2.
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.		 [latex]V=s^3[/latex]
Step 5. Solve the equation. [latex]\begin{array}{rl}\hfill V &= 2^3\hfill \\ \hfill V &= 8\hfill \end{array}[/latex]
Step 6. Check: Check that you did the calculations
correctly.
Step 7. Answer the question. The volume is 8 cubic inches.
Show Solution to (b)
Explanation Steps
Step 1. Use the same figure from above. A cube with length, width, and height labeled as 2.
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.		 [latex]S=6s^2[/latex]
Step 5. Solve the equation. [latex]\begin{array}{rl}\hfill S &= 6\cdot 2^2\hfill \\ \hfill S &= 24\hfill \end{array}[/latex]
Step 6. Check: The check is left to you.
Step 7. Answer the question. The surface area is 24 square inches.

Try It

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

Show Solution

(a) 64 cubic feet; (b) 96 square feet

Try It

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.

Show Solution

(a) 4,096 cubic inches; (b) 1,536 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 [latex]\pi[/latex] with 3.14.

Volume and Surface Area of a Sphere

For a sphere with radius [latex]r[/latex]:

A sphere with radius r. Volume: V equals four-thirds times pi times r cubed. Below that is Surface Area: S equals 4 times pi times r squared.
Figure 7

Volume and Surface Area of a Sphere

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

Show Solution to (a)
Explanation Steps
Step 1. Read the problem. Draw the figure and label
 it with the given information.		 A sphere labeled with a radius of 6.
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.		 [latex]V=\frac{4}{3}\pi r^3[/latex]
Step 5. Solve. [latex]\begin{array}{rl}\hfill V &\approx\frac{4}{3}(3.14)6^3\hfill \\ \hfill V &\approx 904.32\hfill \end{array}[/latex]
Step 6. Check: Double-check your math on a calculator.
Step 7. Answer the question. The volume is approximately 904.32 cubic inches.
Show Solution
Explanation Steps
Step 1. Use the same figure from above. A sphere labeled with a radius of 6.
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.		 [latex]S=4\pi r^2[/latex]
Step 5. Solve. [latex]\begin{array}{rl}\hfill S &\approx 4(3.14)6^2\hfill \\ \hfill S &\approx 452.16\hfill \end{array}[/latex]
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

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

Show Solution

(a) 113.04 cubic centimeters; (b) 113.04 square centimeters

Try It

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

Show Solution

(a) 4.19 cubic feet; (b) 12.56 square feet

Finding Volume and Surface Area of a Sphere

A globe of Earth is in the shape of a sphere with radius 14 inches. Find its (a) volume and (b) surface area. Round the answer to the nearest hundredth.

Show Solution to (a)
Explanation Steps
Step 1. Read the problem. Draw a figure with the
 given information and label it.		 A globe labeled with a radius of 14.
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 ππ)		 [latex]\begin{array}{rl}\hfill V &=\frac{4}{3}\pi r^3\hfill \\ \hfill V &\approx\frac{4}{3}(3.14)14^3\hfill \end{array}[/latex]
Step 5. Solve. [latex]V\approx 11{,}488.21[/latex]
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.
Show Solution to (b)
Explanation Steps
Step 1. Use the figure from above. A globe labeled with a radius of 14.
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 ππ)		 [latex]\begin{array}{rl}\hfill S &= 4\pi r^2\hfill \\ \hfill S &\approx 4(3.14)14^2\hfill \end{array}[/latex]
Step 5. Solve. [latex]S\approx 2{,}461.76[/latex]
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

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

Show Solution

(a) 3,052.08 cubic inches; (b) 1017.36 square inches

Try It

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

Show Solution

(a) 42.39 cubic feet; (b) 28.26 square feet

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, [latex]h[/latex], of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, [latex]h[/latex], will be perpendicular to the bases.
A cylinder with radius r, and h height.
Figure 8
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, [latex]V = Bh[/latex], can also be used to find the volume of a cylinder.

For the rectangular solid, the area of the base, [latex]B[/latex], is the area of the rectangular base, length × width. For a cylinder, the area of the base, [latex]B[/latex], is the area of its circular base, [latex]\pi r^2[/latex]. 

Figure 9 compares how the formula [latex]V=Bh[/latex] is used for rectangular solids and cylinders.

(a) Rectangular solid with sides L, W, H; steps show V=Bh=lw·h=lwh. (b) Cylinder with radius r, height h; V=Bh=πr²h.
Figure 9
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 10.
Cylindrical can of green beans with height h; top and bottom shown as circles. The removed label is a rectangle with length 2πr, height h, width w; C=2πr.
Figure 10
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 [latex]L[/latex] of the rectangular label. The height of the cylinder is the width, [latex]w[/latex], of the rectangular label. So the area of the label can be represented as


The top line says A equals l times red w. Below the l is 2 times pi times r. Below the w is a red h.

To find the total surface area of the cylinder, we add the areas of the two circles to the area of the rectangle.

A cylinder is broken down into one rectangle with 2 circles for cylinder’s ends; shows Surface area = πr² + πr² + 2πrh = 2πr² + 2πrh.
Figure 11

The surface area of a cylinder with radius [latex]r[/latex] and height [latex]h[/latex] is [latex]S=2\pi r^2+2\pi r h[/latex].

Volume and Surface Area of a Cylinder

For a cylinder with radius [latex]r[/latex] and height [latex]h[/latex]:

Cylinder with height h and radius r; volume shown as V=πr²h or Bh, and surface area as S=2πr²+2πrh.
Figure 12

Finding Volume and Surface Area of a Cylinder

A cylinder has height 5 inches and radius 3 inches. Find the (a) volume and (b) surface area.

Show Solution
Explanation Steps
Step 1. Read the problem. Draw the figure and label
 it with the given information.		 A right circular cylinder is shown to have a radius of 3 and a height of 5.
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 ππ)		 [latex]\begin{array}{rl}\hfill V&=\pi r^2h\hfill \\ \hfill V &\approx (3.14)3^2\cdot 5\hfill \end{array}[/latex]
Step 5. Solve. [latex]V\approx 141.3[/latex]
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.
Show Solution
Explanation Steps
Step 1. Use the picture from above. A right circular cylinder is shown to have a radius of 3 and a height of 5.
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 ππ)		 [latex]\begin{array}{rl} \hfill S &= 2\pi r^2+2\pi rh\hfill \\ \hfill S &\approx 2(3.14)3^2+2(3.14)(3)5\hfill \end{array}[/latex]
Step 5. Solve. [latex]S\approx 150.72[/latex]
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

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

Show Solution

(a) 351.68 cubic centimeters; (b) 276.32 square centimeters

Try It

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

Show Solution

(a) 100.48 cubic feet; (b) 125.6 square feet

Finding Volume and Surface Area of a Cylinder

Find the (a) volume and (b) 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.

Show Solution
Explanation Steps
Step 1. Read the problem. Draw the figure and
label it with the given information.		 An illustration of a soda can showing a radius of 4 and a height of 13.
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)		 [latex]\begin{array}{rl}\hfill V &=\pi r^2h\hfill \\ \hfill V &\approx (3.14)4^2\cdot 13\hfill \end{array}[/latex]
Step 5. Solve. [latex]V\approx 653.12[/latex]
Step 6. Check: We leave it to you to check.
Step 7. Answer the question. The volume is approximately 653.12 cubic centimeters.
Show Solution
Explanation Steps
Step 1. Use the same image from before. An illustration of a soda can showing a radius of 4 and a height of 13.
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 ππ)		 [latex]\begin{array}{rl}\hfill S &= 2\pi r^2 +2\pi rh \hfill \\ \hfill S &\approx 2(3.14)4^2+2(3.14)(4)13\hfill \end{array}[/latex]
Step 5. Solve. [latex]S\approx 427.04[/latex]
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

Find the (a) volume and (b) surface area of a can of paint with radius 8 centimeters and height 19 centimeters. Assume the can is shaped exactly like a cylinder.

Show Solution

(a) 3,818.24 cubic centimeters; (b) 1,356.48 square centimeters

Try It

Find the (a) volume and (b) surface area of a cylindrical drum with radius 2.7 feet and height 4 feet. Assume the drum is shaped exactly like a cylinder.

Show Solution

(a) 91.6 cubic feet; (b) 113.6 square feet

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 13.
A cone with top labeled vertex. The height is h. The radius of the base is r.
Figure 13 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 [latex]V=\pi r^2h[/latex]. We can think of a cone as part of a cylinder. Figure 14 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.
A cone drawn inside a cylinder.
Figure 14 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

[latex]V=\frac{1}{3}Bh[/latex]

Since the base of a cone is a circle, we can substitute the formula of area of a circle, [latex]B=\pi r^2[/latex], to get the formula for volume of a cone.

[latex]V=\frac{1}{3}\pi r^2h[/latex]

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 [latex]r[/latex] and height [latex]h[/latex]:

A cone with height h, the radius of the base r. Beside this is Volume: V equals one-third times pi times r squared times h.
Figure 15

Mathispower4u. Ex: Determine Volume of a Cone. YouTube. https://www.youtube.com/watch?v=7Y0ZMnCcVGs&t=1s

Finding Volume of a Cone

Find the volume of a cone with height 6 inches and radius of its base 2 inches.

Show Solution
Explanation Steps
Step 1. Read the problem. Draw the figure and label it
 with the given information.		 A cone is labeled with a base radius of 2 and a height of 6.
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 ππ)		 [latex]\begin{array}{rl}\hfill V &=\frac{1}{3}\pi r^2h\hfill \\ \hfill V &\approx \frac{1}{3}(3.14)(2)^2(6)\hfill \end{array}[/latex]
Step 5. Solve. [latex]V\approx 25.12[/latex]
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

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

Show Solution

65.94 cubic inches

Try It

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

Show Solution

235.5 cubic centimeters

Finding Volume of a Cone

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.

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Explanation Steps
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. An inverted cone is shown with a base radius of 5 and a height of 8.
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 [latex]\pi[/latex], and notice that we were given the distance across the circle, which is its diameter. The radius is 2.5 inches.) [latex]\begin{array}{rl}\hfill V &=\frac{1}{3}\pi r^2h\hfill \\ \hfill V &\approx \frac{1}{3}(3.14)(2.5)^2(8)\hfill \end{array}[/latex]
Step 5. Solve. [latex]V\approx 52.33[/latex]
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

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.

Show Solution

678.24 cubic inches

Try It

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.

Show Solution

128.22 cubic inches

Key Concepts

Volume and Surface Area of a Rectangular Solid

  • [latex]V = L \cdot W \cdot H[/latex]
  • [latex]A = 2LW + 2LH + 2WH[/latex]

Volume and Surface Area of a Cube

  • [latex]V = s^3[/latex]
  • [latex]A = 6s^2[/latex]

Volume and Surface Area of a Sphere

  • [latex]V = \frac{4}{3} \pi r^3[/latex]
  • [latex]A = 4 \pi r^2[/latex]

Volume and Surface Area of a Cylinder

  • [latex]V = \pi r^2 h[/latex]
  • [latex]A = 2\pi r (r + h)[/latex] or [latex]A = 2\pi r^2 + 2\pi rh[/latex]

Volume of a Cone

  • [latex]V = \frac{1}{3} \pi r^2 h[/latex]

Summary of Geometry Formulas

The following charts summarize all of the formulas covered in this chapter.Table summarizing chapter formulas with images: angles, triangles, trapezoid, similar triangles, sphere, circle, cylinder, cone, .
Table shows more chapter formulas: rectangular solid (V=LWH, S=2LH+2LW+2WH), rectangle (P=2L+2W, A=LW), and cube (V=s³, S=6s²) with images.

Section Exercises

Find Volume and Surface Area of Rectangular Solids

In the following exercises, find (a) the volume and (b) the surface area of the rectangular solid with the given dimensions.

1. Length 2 meters, width 1.5 meters, height 3 meters

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(a) 9 cubic meters; (b) 27 square meters

2. Length 5 feet, width 8 feet, height 2.5 feet

3. Length 3.5 yards, width 2.1 yards, height 2.4 yards

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(a) 17.64 cubic yards; (b) 41.58 square yards

4. Length 8.8 centimeters, width 6.5 centimeters, height 4.2 centimeters

 

In the following exercises, solve.

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

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(a) 1,024 cubic feet; (b) 640 square feet

6. Gift box A rectangular gift box has length 26 inches, width 16 inches, and height 4 inches. Find its (a) volume and (b) surface area.

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

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(a) 3,350.49 cubic cm; (b) 1,622.42 square cm

8. Shipping container A rectangular shipping container has length 22.8 feet, width 8.5 feet, and height 8.2 feet. Find its (a) volume and (b) surface area.

 

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

9. 5 centimeters

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(a) 125 cubic centimeters; (b) 150 square centimeters

10.6 inches

11. 10.4 feet

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(a) 1,124.9 cubic feet; (b) 648.96 square feet

12. 12.5 meters

 

In the following exercises, solve.

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

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(a) 262,144 cubic feet; (b) 24,576 square feet

14. Museum A cube-shaped museum has sides 45 meters long. Find its (a) volume and (b) surface area.

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

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(a) 21.952 cubic meters; (b) 47.04 square meters

16. Tissue box A box of tissues is a cube with sides 4.5 inches long. Find its (a) volume and (b) surface area.

 

Find the Volume and Surface Area of Spheres

In the following exercises, find (a) the volume and (b) the surface area of the sphere with the given radius. Round answers to the nearest hundredth.

17. 3 centimeters

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(a) 113.04 cubic centimeters; (b) 113.04 square centimeters

18. 9 inches

19. 7.5 feet

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(a) 1766.25 cubic feet; (b) 706.5 square feet

20. 2.1 yards

 

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

21. Exercise ball An exercise ball has a radius of 15 inches. Find its (a) volume and (b) surface area.

Show Solution

(a) 14,130 cubic inches; (b) 2,826 square inches

22. Balloon ride The Great Park Balloon is a big orange sphere with a radius of 36 feet. Find its (a) volume and (b) surface area.

23. Golf ball A golf ball has a radius of 4.5 centimeters. Find its (a) volume and (b) surface area.

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(a) 381.51 cubic centimeters; (b) 254.34 square centimeters

24. Baseball A baseball has a radius of 2.9 inches. Find its (a) volume and (b) surface area.

 

Find the Volume and Surface Area of a Cylinder

In the following exercises, find (a) the volume and (b) the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth.

25. Radius 3 feet, height 9 feet

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(a) 254.34 cubic feet; (b) 226.08 square feet

26. Radius 5 centimeters, height 15 centimeters

27. Radius 1.5 meters, height 4.2 meters

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(a) 29.673 cubic meters; (b) 53.694 square meters

28. Radius 1.3 yards, height 2.8 yards

 

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

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

Show Solution

(a) 1020.5 cubic cm; (b) 565.2 square cm

30. Snack pack A snack pack of cookies is shaped like a cylinder with radius 4 cm and height 3 cm. Find its (a) volume and (b) surface area.

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

Show Solution

(a) 678.24 cubic inches; (b) 508.68 square inches

32. Architecture A cylindrical column has a diameter of 8 feet and a height of 28 feet. Find its (a) volume and (b) surface area.

 

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.

33. Height 9 feet and radius 2 feet

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37.68 cubic feet

34. Height 8 inches and radius 6 inches.

35. Height 12.4 centimeters and radius 5 cm

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397.7 cubic centimeters

36. Height 15.2 meters and radius 4 meters

 

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

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

Show Solution

261.67 cubic feet

38. Popcorn cup What is the volume of a cone-shaped popcorn cup that is 8 inches tall and 6 inches across at the base?

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

Show Solution

64,108.3 cubic feet

40. Sand pile What is the volume of a cone-shaped pile of sand that is 12 meters tall and 30 meters across at the base?

Everyday Math

41. 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.

    A cone with a dark dotted line at the top indicating a smaller cone.
Show Solution

(a) 31.4 cubic feet; (b) 2.6 cubic feet; (c) 28.8 cubic feet

42. Ice cream cones A regular ice cream cone is 4 inches tall and has a diameter of 2.5  inches. A waffle cone is 7 inches tall and has a diameter of 3.25 inches. To the nearest hundredth,

  1. Find the volume of the regular ice cream cone.
  2. Find the volume of the waffle cone.
  3. How much more ice cream fits in the waffle cone compared to the regular cone?

Writing Exercises

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

Show Solution

A cone fits inside a cylinder with the same height and base radius, so its volume is smaller.

44. Which has a larger volume, a cube of sides of 8 feet or a sphere with a diameter of 8 feet? Explain your reasoning.

Glossary

cone

A cone is a solid figure with one circular base and a vertex.

cube

A cube is a rectangular solid whose length, width, and height are equal.

cylinder

A cylinder is a solid figure with two parallel circles of the same size at the top and bottom.

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