2. Geometry Surface Area and Volume of Spheres

3. October 18, 2015 Goals Find the surface area of spheres. Find the volume of spheres. Solve problems using area and volume.

4. October 18, 2015 Sphere radius Great Circle (Divides the sphere into two halves.) Sphere Demo The set of points in space that are equidistant from the same point, the center.

5. October 18, 2015 Hemisphere r Half of a sphere.

6. October 18, 2015 Sphere Formulas r

7. October 18, 2015 Using a Calculator You may find it easier to use the formula for volume in this form:

8. October 18, 2015 Example 2 Find the Surface Area and the Volume of a sphere with a radius of 2.

9. October 18, 2015 Your Turn Find the surface area and volume. 7 in.

10. October 18, 2015 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.

11. October 18, 2015 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.

12. October 18, 2015 Problem 1 Solution The radii are 12 in, 10 in, and 8 in. 13,571.68 cu in

13. October 18, 2015 Or, for easier calculation…

14. October 18, 2015 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.

15. October 18, 2015 Problem 2 Solution 50 18 Volume of Cylinder V =  r 2 h V =  (9 2 )(50) V =   81  50 V = 4050  V  12723.5 cu. ft. 9

16. October 18, 2015 Problem 2 Solution Volume of Hemisphere 50 18 9 This is the volume of a sphere. The volume of the hemisphere is half of this value, which is 1526.8 cu. ft.

17. October 18, 2015 Problem 2 Solution Volume of Cylinder 12723.5 Volume of Hemisphere 1526.8 Total Volume 12723.5 + 1526.8 = 14250.3 cu. ft. 50 18 9

18. October 18, 2015 Problem 2 Extension Total Volume =14250.3 cu. ft. One bushel contains 1.244 cubic feet. How many bushels are in the silo? 14250.3  1.244 = 11455.2 bushels

19. October 18, 2015 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? Skip

20. October 18, 2015 Problem 3 Solution 6 6 6 Volume of the Cube: 6  6  6 = 216 cu in Radius of the Sphere: 3 in. Volume of the Sphere: 3

21. October 18, 2015 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 3

22. October 18, 2015 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 cylindrical 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? Skip

23. October 18, 2015 Problem 4 Solution Flask Volume: Diameter = 4 inches Radius = 2 inches

24. October 18, 2015 Problem 4 Solution Cup Volume: Diameter = 3.5 inches Radius = 1.75 inches Height = 3.5 inches 33.5 cu in

25. October 18, 2015 Problem 4 Solution The flask holds 33.5 cu in. The cup holds 33.7 cu in. Yes, the potion fits into the cup. 33.5 cu in 33.7 cu in

26. October 18, 2015 Problem 5 The surface area of a sphere is 300 cm 2. Find: 1. Its radius, 2. The circumference of a great circle, 3. Its volume. Skip

27. October 18, 2015 Problem 5 Solutions

28. October 18, 2015 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.

29. October 18, 2015 Formulas to Know r

30. October 18, 2015 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? Skip

31. October 18, 2015 Last Problem Solution

32. October 18, 2015 Practice Problems