1. 12.6 Surface Area and Volume of Spheres
Geometry
12.6 Surface Area and Volume of Spheres

2. Geometry 11.7 Surface Area and Volume of Spheres
11.7 Essential Question
How do you find the surface area and volume of a sphere?
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Geometry 11.7 Surface Area and Volume of Spheres

3. Geometry 11.7 Surface Area and Volume of Spheres
Goals
Find the surface area of spheres.
Find the volume of spheres.
Solve problems using area and volume.
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Geometry 11.7 Surface Area and Volume of Spheres

4. Geometry 11.7 Surface Area and Volume of Spheres
The set of points in space that are equidistant from the same point, the center.
radius
Great Circle
(Divides the sphere into two halves.)
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Geometry 11.7 Surface Area and Volume of Spheres

5. Geometry 11.7 Surface Area and Volume of Spheres
Hemisphere
Half of a sphere. r
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Geometry 11.7 Surface Area and Volume of Spheres

6. Geometry 11.7 Surface Area and Volume of Spheres
Sphere Formulas r
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Geometry 11.7 Surface Area and Volume of Spheres

7. Geometry 11.7 Surface Area and Volume of Spheres
Using a Calculator
You may find it easier to use the formula for volume in this form:
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Geometry 11.7 Surface Area and Volume of Spheres

8. Geometry 11.7 Surface Area and Volume of Spheres
Example 1
Find the Surface Area and the Volume of a sphere with a radius of 2. 2
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Geometry 11.7 Surface Area and Volume of Spheres

9. Your Turn Find the surface area and volume.
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Geometry 11.7 Surface Area and Volume of Spheres

10. Geometry 11.7 Surface Area and Volume of Spheres
Problem 1
A snowman is made with three spheres. The largest has a diameter of 24 inches, the next largest has a diameter of 20 inches, and the smallest has a diameter of 16 inches. Find the volume of the snowman.
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Geometry 11.7 Surface Area and Volume of Spheres

11. Geometry 11.7 Surface Area and Volume of Spheres
Problem 1 Solution
A snowman is made with three spheres. The largest has a diameter of 24 inches, the next largest has a diameter of 20 inches, and the smallest has a diameter of 16 inches. Find the volume of the snowman.
The radii are 12 in, 10 in, and 8 in.
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Geometry 11.7 Surface Area and Volume of Spheres

12. Geometry 11.7 Surface Area and Volume of Spheres
Problem 1 Solution
The radii are 12 in, 10 in, and 8 in.
13, cu in
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Geometry 11.7 Surface Area and Volume of Spheres

13. Or, for easier calculation…
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Geometry 11.7 Surface Area and Volume of Spheres

14. Geometry 11.7 Surface Area and Volume of Spheres
Problem 2
This is a grain silo, as found on many farms. They are used to store feed grain and other materials. They are usually cylindrical with a hemispherical top.
Assume that the concrete part has a height of 50 feet, and the diameter of the cylinder is 18 feet.
Find the volume of the silo.
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Geometry 11.7 Surface Area and Volume of Spheres

15. Geometry 11.7 Surface Area and Volume of Spheres
Problem 2 Solution
Volume of Cylinder
V = r2h
V = (92)(50)
V =   81  50
V = 4050
V  cu. ft. 18 50 9
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Geometry 11.7 Surface Area and Volume of Spheres

16. Geometry 11.7 Surface Area and Volume of Spheres
Problem 2 Solution
Volume of Hemisphere 18 50 9
This is the volume of a sphere. The volume of the hemisphere is half of this value, which is cu. ft.
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Geometry 11.7 Surface Area and Volume of Spheres

17. Geometry 11.7 Surface Area and Volume of Spheres
Problem 2 Solution
Volume of Cylinder
Volume of Hemisphere
1526.8
Total Volume
= cu. ft. 18 50 9
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Geometry 11.7 Surface Area and Volume of Spheres

18. Geometry 11.7 Surface Area and Volume of Spheres
Problem 2 Extension
Total Volume = cu. ft.
One bushel contains cubic feet. How many bushels are in the silo?
 =
bushels
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Geometry 11.7 Surface Area and Volume of Spheres

19. Geometry 11.7 Surface Area and Volume of Spheres
Problem 3
A sphere is inscribed inside a cube which measures 6 in. on a side. What is the ratio of the volume of the sphere to the volume of the cube?
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Geometry 11.7 Surface Area and Volume of Spheres

20. Geometry 11.7 Surface Area and Volume of Spheres
Problem 3 Solution
Volume of the Cube:
6  6  6 = 216 cu in
Radius of the Sphere:
3 in.
Volume of the Sphere: 6 6 3 6
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Geometry 11.7 Surface Area and Volume of Spheres

21. Problem 3 Solution 6 6 6 Volume of the Cube: 216 cu in
Volume of the Sphere:
36
Ratio of Volume of Sphere to Volume of Cube 6 6 3 6
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Geometry 11.7 Surface Area and Volume of Spheres

22. Geometry 11.7 Surface Area and Volume of Spheres
Problem 4
A mad scientist makes a potion in a full spherical flask which has a diameter of 4 inches. To drink it, he pours it into a cup with a diameter of 3.5 inches and is 3.5 inches high. Will the potion fit into the cup? If not, how much is left in the flask?
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Geometry 11.7 Surface Area and Volume of Spheres

23. Geometry 11.7 Surface Area and Volume of Spheres
Problem 4 Solution
Flask Volume:
Diameter = 4 inches
Radius = 2 inches
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Geometry 11.7 Surface Area and Volume of Spheres

24. Geometry 11.7 Surface Area and Volume of Spheres
Problem 4 Solution
33.5 cu in
Cup Volume:
Diameter = 3.5 inches
Radius = 1.75 inches
Height = 3.5 inches
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Geometry 11.7 Surface Area and Volume of Spheres

25. Geometry 11.7 Surface Area and Volume of Spheres
Problem 4 Solution
33.5 cu in
33.7 cu in
The flask holds 33.5 cu in.
The cup holds 33.7 cu in.
Yes, the potion fits into the cup.
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Geometry 11.7 Surface Area and Volume of Spheres

26. Geometry 11.7 Surface Area and Volume of Spheres
Problem 5
The surface area of a sphere is 300 cm2.
Find:
Its radius,
The circumference of a great circle,
Its volume.
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Geometry 11.7 Surface Area and Volume of Spheres

27. Geometry 11.7 Surface Area and Volume of Spheres
Problem 5 Solutions
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Geometry 11.7 Surface Area and Volume of Spheres

28. Geometry 11.7 Surface Area and Volume of Spheres
Last Problem
On a far off planet, Zenu was examining his next target, Earth. The radius of the Earth is 3963 miles. What is the volume of material that will be blown into space?
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Geometry 11.7 Surface Area and Volume of Spheres

29. Geometry 11.7 Surface Area and Volume of Spheres
Last Problem Solution
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Geometry 11.7 Surface Area and Volume of Spheres

30. Geometry 11.7 Surface Area and Volume of Spheres
Summary
A sphere is the set of points in space equidistant from a center.
A hemisphere is half of a sphere.
A great circle is the largest circle that can be drawn on a sphere. The diameter of a great circle equals the diameter of the sphere.
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Geometry 11.7 Surface Area and Volume of Spheres

31. Geometry 11.7 Surface Area and Volume of Spheres
Formulas to Know r
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Geometry 11.7 Surface Area and Volume of Spheres

32. Geometry 11.7 Surface Area and Volume of Spheres
Homework
Page 762
Problems 10 – 17, 20 – 29 all
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Geometry 11.7 Surface Area and Volume of Spheres