Volume of Spheres. A geometric plane passing through the center of a sphere divides it into. into the Northern Hemisphere and the Southern Hemisphere.
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1 9.6 Surface Area and Volume of Spheres Goal Find surface areas and volumes of spheres. Key Words sphere hemisphere A globe is an example of a sphere. A sphere is the set of all points in space that are the same distance from a point, the center of the sphere. A geometric plane passing through the center of a sphere divides it into two hemispheres. The globe is divided into the Northern Hemisphere and the Southern Hemisphere. Northern Hemisphere Southern Hemisphere The globe is divided into two hemispheres. SURFACE AREA OF A SPHERE Words Surface area 4π(radius) 2 Symbols S 4πr 2 center radius Visualize It! To sketch a sphere, draw a circle and its center. Then draw an oval to give the sphere dimension. EXAMPLE 1 F d th S rfa Area of a Sph re Find the surface area of the sphere. Round your answer to the nearest whole number. a. b. 10 cm Solution a. The radius is 8 inches, b. The diameter is 10 cm, so the so r 8. radius is So, r 5. 2 S 4πr 2 S 4πr 2 4 pπp8 2 4 pπp The surface area is about The surface area is about 804 square inches. 14 square centimeters. 9.6 Surface Area and Volume of Spheres 517
2 Find the Surface Area of a Sphere Find the surface area of the sphere. Round your answer to the nearest whole number ft 1 VOLUME OF A SPHERE Words Volume 4 π(radius) Symbols V 4 πr Student Help I CLASSZONE.COM MORE EXAMPLES More examples at classzone.com EXAMPLE 2 Find the Volume of a Sphere Find the volume of the sphere or hemisphere. Round your answer to the nearest whole number. a. b. Solution a. V 4 πr Write the formula for volume of a sphere. 4 pπp2 Substitute 2 for r. 2 π Simplify. 2 2 p 2 p ANSWER Multiply. The volume is about 4 cubic feet. b. A hemisphere has half the volume of a sphere. V πr Write the formula for 1 the volume of a sphere p 4 pπp5 Substitute 5 for r π Simplify. 5 5 p 5 p ANSWER Multiply. The volume is about 262 cubic inches. 518 Chapter 9 Surface Area and Volume
3 EXAMPLE Find the Volume of a Sphere Estimate the volume of air in a beach ball that has a 12 inch diameter. Round your answer to the nearest whole number. Solution V 4 πr Write volume formula. 4 pπp6 Substitute for r. 288π 905 Simplify. Multiply. ANSWER The volume of air in the ball is about 905 cubic inches. Find the Volume of a Sphere Find the volume to the nearest whole number cm 18 ft 9.6 Exercises Guided Practice Vocabulary Check Skill Check 1. Explain the difference between a sphere and a hemisphere. Find the surface area to the nearest whole number in. ft 9 m Find the volume to the nearest whole number ft 11 cm 20 yd 9.6 Surface Area and Volume of Spheres 519
4 Practice and Applications Extra Practice See p Find Surface Area of a Sphere Find the surface area of the sphere cm 16 in. 0 m 11. Error Analysis Bob is asked to find the surface area of a sphere with a diameter of 10 millimeters. Explain and correct his error(s). V πr 2 π(10) 2 100π 14 mm 10 mm Sports In Exercises 12 17, estimate the surface area of the ball. 12. Soccer ball 1. Tennis ball 14. Bowling ball r. cm r 4. in. r 10.9 cm 15. Golf ball 16. Basketball 17. Softball Homework Help Example 1: Exs Example 2: Exs. 2 28, 7 9 Example : Exs. 2 28, 7 9 d 1.7 in. You be the Judge d 9.5 in. 18. Julie thinks that if you double the radius of the sphere shown at the right, the surface area will double. Is she right Explain your reasoning. d 9.6 cm cm 520 Chapter 9 Surface Area and Volume
5 Astronomy Astronomy In Exercises 19 22, use the information about Earth and its moon given in the photo. 19. Find the surface area of Earth. 20. Find the surface area of Earth s moon. Moon s radius: about 1080 miles 21. Compare the surface areas of Earth and its moon. 22. About 70% of Earth s surface is water. How many square miles of water are on Earth s surface Earth s radius: about 960 miles Finding Volume of a Sphere Find the volume of the sphere. ASTRONOMERS study the planets, stars, and solar system. Powerful telescopes are used to collect information about astronomical objects. Application Links CLASSZONE.COM m 4 ft yd 14 ft 10 cm 7 in. Technology Use formulas to create a spreadsheet like the one shown. Then answer Exercises A Radius, r Comparing Spheres C Surface area of new sphere Surface area of original sphere B Surface area, 4πr How many times greater is the surface area of a sphere if the radius is doubled tripled quadrupled 0. Explain why the surface area changes by a greater amount than the radius. 1. How many times greater do you think the volume of a sphere will be if the radius is doubled tripled 2. Create a spreadsheet for the volume of a sphere. Then answer Exercises 29 and 0 for the volume of a sphere. 9.6 Surface Area and Volume of Spheres 521
6 Student Help LOOK BACK See pp for more information about The Rose Center for Earth and Space. Spheres in Architecture In Exercises 6, refer to the information below about The Rose Center for Earth and Space at New York City s American Museum of Natural History. The sphere has a diameter of 87 feet. The glass cube surrounding the sphere is 95 feet long on each edge.. Find the surface area of the sphere. 4. Find the volume of the sphere. 5. Find the volume of the glass cube. 6. Find the approximate amount of glass used to make the cube. (Hint: Do not include the ground or roof in your calculations.) Finding Volume of a Hemisphere Find the volume of the hemisphere cm 15 m 18 in. Composite Solids Find the volume of the solid. Round your answer to the nearest whole number cm cm 6 ft 12 ft 9 in. 5 in. IStudent Help I CLASSZONE.COM HOMEWORK HELP Extra help with problem solving in Exs is at classzone.com Standardized Test Practice Architecture The entrance to the Civil Rights Institute in Birmingham, Alabama, includes a hemisphere that has a radius of 25. feet. 4. Find the volume of the hemisphere. 44. Find the surface area of the hemisphere, not including its base. 45. The walls of the hemisphere are 1. feet thick. So, the rounded surface inside the building is a hemisphere with a radius of 24 feet. Find its surface area, not including its base. 46. Multiple Choice What is the approximate surface area of the sphere shown A 217 in. 2 B 4287 in. 2 C 12,861 in. 2 D 17,149 in. 2 2 in. 25. ft 522 Chapter 9 Surface Area and Volume
7 Mixed Review Surface Area Find the surface area of the solid. If necessary, round your answer to the nearest whole number. (Lessons 9.2, 9.) 47. A cone has a height of 12 meters and a base radius of meters. 48. A pyramid has a slant height of feet and a square base that measures 4 feet on a side. 49. A cylinder has a radius of 9 centimeters and a height of 9 centimeters. Simplifying Radicals Evaluate. Give the exact value if possible. Otherwise, approximate to the nearest tenth. (Skills Review, p. 668) Algebra Skills Using Formulas Find the missing length using the given information. (Skills Review, p. 674) 58. A rectangle is 6 feet wide and 11 feet long. Find the perimeter. 59. A square has an area of 100 square inches. Find the perimeter. 60. Find the width of a rectangle with a length of 8 meters and an area of 40 square meters. 61. The perimeter of a square is 44 yards. Find the side length. Quiz 2 Find the volume of the solid. If necessary, round your answer to the nearest whole number. (Lessons 9.4, 9.5, 9.6) ft 10 ft 9 in. 7 ft 10 m 7 m in. 12 ft 10 ft 10 ft 16 m 5 in. 7. Sketch a cylinder with a radius of 4 inches and a height of 4 inches. Then find its volume. (Lesson 9.4) 8. Sketch a sphere with a radius of 9 centimeters. Then find its surface area. (Lesson 9.6) 9.6 Surface Area and Volume of Spheres 52
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