11.6: Surface Area and Volume of Spheres
Learning Objectives
- Find the surface area of a sphere.
- Find the volume of a sphere.
Review Queue
- List three spheres you would see in real life.
- Find the area of a circle with a 6 cm radius.
- Find the volume of a cylinder with the circle from #2 as the base and a height of 5 cm.
Know What? A regulation bowling ball is a sphere with a circumference of 27 inches. Find the radius of a bowling ball, its surface area and volume. You may assume the bowling ball does not have any finger holes. Round your answers to the nearest hundredth.
Defining a Sphere
A sphere is the last of the three-dimensional shapes that we will find the surface area and volume of. Think of a sphere as a three-dimensional circle.
Sphere: The set of all points, in three-dimensional space, which are equidistant from a point.
The radius has an endpoint on the sphere and the other endpoint is the center.
The diameter must contain the center.
Great Circle: A cross section of a sphere that contains the diameter.
A great circle is the largest circle cross section in a sphere. The circumference of a sphere is the circumference of a great circle.
Every great circle divides a sphere into two congruent hemispheres.
Example 1: The circumference of a sphere is \begin{align*}26 \pi \ feet\end{align*}
Solution: The circumference is referring to the circumference of a great circle. Use \begin{align*}C = 2 \pi r\end{align*}
\begin{align*}2 \pi r &= 26\pi\\
r &= 13 \ ft.\end{align*}
Surface Area of a Sphere
The best way to understand the surface area of a sphere is to watch the link by Russell Knightley, http://www.rkm.com.au/ANIMATIONS/animation-Sphere-Surface-Area-Derivation.html.
Surface Area of a Sphere: \begin{align*}SA=4 \pi r^2\end{align*}
Example 2: Find the surface area of a sphere with a radius of 14 feet.
Solution:
\begin{align*}SA &= 4\pi (14)^2\\
&= 784 \pi \ ft^2\end{align*}
Example 3: Find the surface area of the figure below.
Solution: Be careful when finding the surface area of a hemisphere because you need to include the area of the base.
\begin{align*}SA &= \pi r^2+\frac{1}{2} 4 \pi r^2\\
&= \pi (6^2 )+2 \pi (6^2)\\
&= 36 \pi +72 \pi =108 \pi \ cm^2\end{align*}
Example 4: The surface area of a sphere is \begin{align*}100 \pi \ in^2\end{align*}
Solution:
\begin{align*}SA &= 4 \pi r^2\\
100 \pi &= 4 \pi r^2\\
25 &= r^2\\
5 &= r\end{align*}
Example 5: Find the surface area of the following solid.
Solution: This solid is a cylinder with a hemisphere on top. It is one solid, so do not include the bottom of the hemisphere or the top of the cylinder.
\begin{align*}SA &=LA_{cylinder}+LA_{hemisphere}+A_{base \ circle}\\
&= \pi rh+\frac{1}{2} 4 \pi r^2+\pi r^2\\
&= \pi (6)(13)+2 \pi 6^2+\pi 6^2\\
&= 78 \pi +72 \pi +36 \pi\\
&= 186 \pi \ in^2 \qquad \qquad “LA” \ \text{stands for} \ lateral \ area.\end{align*}
Volume of a Sphere
To see an animation of the volume of a sphere, see http://www.rkm.com.au/ANIMATIONS/animation-Sphere-Volume-Derivation.html by Russell Knightley.
Volume of a Sphere: \begin{align*}V=\frac{4}{3} \pi r^3\end{align*}
Example 6: Find the volume of a sphere with a radius of 9 m.
Solution:
\begin{align*}V &= \frac{4}{3} \pi 6^3\\
&= \frac{4}{3} \pi (216)\\
&= 288 \pi \ m^3\end{align*}
Example 7: A sphere has a volume of \begin{align*}14137.167 \ ft^3\end{align*}
Solution:
\begin{align*}V &= \frac{4}{3} \pi r^3\\
14137.167 &= \frac{4}{3} \pi r^3\\
\frac{3}{4 \pi} \cdot 14137.167 &= r^3\\
3375 &= r^3\end{align*}
At this point, you will need to take the cubed root of 3375. Ask your teacher how to do this on your calculator.
\begin{align*}\sqrt[3]{3375}=15=r\end{align*}
Example 8: Find the volume of the following solid.
Solution:
\begin{align*}V_{cylinder} &= \pi 6^2 (13)=78 \pi\\ V_{hemisphere} &= \frac{1}{2} \left(\frac{4}{3} \pi 6^3\right)=36 \pi\\ V_{total} &= 78 \pi+36 \pi =114 \pi \ in^3\end{align*}
Know What? Revisited The radius would be \begin{align*}27=2 \pi r\end{align*}, or \begin{align*}r=4.30 \ inches\end{align*}. The surface area would be \begin{align*}4 \pi 4.3^2 \approx 232.35 \ in^2\end{align*}, and the volume would be \begin{align*}\frac{4}{3} \pi 4.3^3 \approx 333.04 \ in^3\end{align*}.
Review Questions
- Questions 1-3 look at the definition of a sphere.
- Questions 4-17 are similar to Examples 1, 2, 4, 6 and 7.
- Questions 18-21 are similar to Example 3 and 5.
- Questions 22-25 are similar to Example 8.
- Question 26 is a challenge.
- Questions 27-29 are similar to Example 8.
- Question 30 analyzes the formula for the surface area of a sphere.
- Are there any cross-sections of a sphere that are not a circle? Explain your answer.
- List all the parts of a sphere that are the same as a circle.
- List any parts of a sphere that a circle does not have.
Find the surface area and volume of a sphere with: (Leave your answer in terms of \begin{align*}\pi\end{align*})
- a radius of 8 in.
- a diameter of 18 cm.
- a radius of 20 ft.
- a diameter of 4 m.
- a radius of 15 ft.
- a diameter of 32 in.
- a circumference of \begin{align*}26 \pi \ cm\end{align*}.
- a circumference of \begin{align*}50 \pi \ yds\end{align*}.
- The surface area of a sphere is \begin{align*}121 \pi \ in^2\end{align*}. What is the radius?
- The volume of a sphere is \begin{align*}47916 \pi \ m^3\end{align*}. What is the radius?
- The surface area of a sphere is \begin{align*}4 \pi \ ft^2\end{align*}. What is the volume?
- The volume of a sphere is \begin{align*}36 \pi \ mi^3\end{align*}. What is the surface area?
- Find the radius of the sphere that has a volume of \begin{align*}335 \ cm^3\end{align*}. Round your answer to the nearest hundredth.
- Find the radius of the sphere that has a surface area \begin{align*}225 \pi \ ft^2\end{align*}.
Find the surface area of the following shapes. Leave your answers in terms of \begin{align*}\pi\end{align*}.
- You may assume the bottom is open.
Find the volume of the following shapes. Round your answers to the nearest hundredth.
- A sphere has a radius of 5 cm. A right cylinder has the same radius and volume. Find the height of the cylinder.
Tennis balls with a 3 inch diameter are sold in cans of three. The can is a cylinder. Round your answers to the nearest hundredth.
- What is the volume of one tennis ball?
- What is the volume of the cylinder?
- Assume the balls touch the can on the sides, top and bottom. What is the volume of the space not occupied by the tennis balls?
- How does the formula of the surface area of a sphere relate to the area of a circle?
Review Queue Answers
- Answers will vary. Possibilities are any type of ball, certain lights, or the 76/Unical orb.
- \begin{align*}36 \pi\end{align*}
- \begin{align*}180 \pi\end{align*}
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