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Geometry

Geometry. Surface Area and Volume of Spheres. Goals. Find the surface area of spheres. Find the volume of spheres. Solve problems using area and volume. Sphere. Sphere Demo. The set of points in space that are equidistant from the same point, the center. radius. Great Circle

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Geometry

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  1. Geometry Surface Area and Volume of Spheres

  2. Goals • Find the surface area of spheres. • Find the volume of spheres. • Solve problems using area and volume.

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

  4. Hemisphere Half of a sphere. r

  5. r Sphere Formulas

  6. Using a Calculator • You may find it easier to use the formula for volume in this form:

  7. 2 Example Find the Surface Area and the Volume of a sphere with a radius of 2.

  8. Your Turn Find the surface area and volume. 7 in.

  9. 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.

  10. 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.

  11. Problem 1 Solution • The radii are 12 in, 10 in, and 8 in. 13,571.68 cu in

  12. Or, for easier calculation…

  13. 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.

  14. Problem 2 Solution Volume of Cylinder V = r2h V = (92)(50) V =   81  50 V = 4050 V  12723.5 cu. ft. 18 50 9

  15. 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 1526.8 cu. ft.

  16. Problem 2 Solution Volume of Cylinder 12723.5 Volume of Hemisphere 1526.8 Total Volume 12723.5 + 1526.8 = 14250.3 cu. ft. 18 50 9

  17. 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

  18. Problem 3 Skip 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?

  19. 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

  20. Problem 3 Solution 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

  21. 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

  22. Problem 4 Solution Flask Volume: Diameter = 4 inches Radius = 2 inches

  23. Problem 4 Solution 33.5 cu in Cup Volume: Diameter = 3.5 inches Radius = 1.75 inches Height = 3.5 inches

  24. 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.

  25. Problem 5 Skip The surface area of a sphere is 300 cm2. Find: • Its radius, • The circumference of a great circle, • Its volume.

  26. Problem 5 Solutions

  27. 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.

  28. r Formulas to Know

  29. Last Problem Skip 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?

  30. Last Problem Solution

  31. Practice Problems

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