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 . What is the radius of the sphere?
Solution: The circumference is referring to the circumference of a great circle. Use .
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: .
Example 2: Find the surface area of a sphere with a radius of 14 feet.
Solution:
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.
Example 4: The surface area of a sphere is . What is the radius?
Solution:
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.
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: .
Example 6: Find the volume of a sphere with a radius of 9 m.
Solution:
Example 7: A sphere has a volume of , what is the radius?
Solution:
At this point, you will need to take the cubed root of 3375. Ask your teacher how to do this on your calculator.
Example 8: Find the volume of the following solid.
Solution:
Know What? Revisited The radius would be , or . The surface area would be , and the volume would be .
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 )
- 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 .
- a circumference of .
- The surface area of a sphere is . What is the radius?
- The volume of a sphere is . What is the radius?
- The surface area of a sphere is . What is the volume?
- The volume of a sphere is . What is the surface area?
- Find the radius of the sphere that has a volume of . Round your answer to the nearest hundredth.
- Find the radius of the sphere that has a surface area .
Find the surface area of the following shapes. Leave your answers in terms of .
- 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.