Section 12.6 Surface Areas and Volumes of Spheres

S=4  r 2 Surface area of a sphere =4  (4.5) 2 Replace r with 4.5. ≈254.5Simplify. Answer:254.5 in 2 Example 1: Find the surface area of the sphere. Round to the nearest tenth and label the units.

A plane can intersect a sphere in a point or in a circle. If the circle contains the center of the sphere, the intersection is called a great circle. The endpoints of a diameter of a great circle are called the poles. Since a great circle has the same center as the sphere and its radii are also radii of the sphere, it is the largest circle that can be drawn on a sphere. A great circle separates a sphere into two congruent halves, called hemispheres.

Find half the area of a sphere with the radius of 3.7 millimeters. Then add the area of the great circle. Example 2: Find the surface area of the hemisphere. Label the units. Surface area of a hemisphere Answer: about  mm 2 Replace r with 3.7. Use a calculator. = 

First, find the radius. The circumference of a great circle is 2  r. So, 2  r = 10  or r = 5. Example 3: Find the surface area of the sphere if the circumference of the great circle is 10π. S=4  r 2 Surface area of a sphere =4  (5) 2 Replace r with 5. =100  Use a calculator. Answer: 100  ft 2

First, find the radius. The area of a great circle is  r 2. So,  r 2 = 220 or r ≈ 8.4. Example 4: Find the surface area of the sphere if the area of the great circle is approximately 220 square meters. Round to the nearest tenth. Answer: about m 2 S=4  r 2 Surface area of a sphere ≈4  (8.4) 2 Replace r with 5. ≈886.7Use a calculator.

Volume of a sphere r = 15 ≈ 14,137.2 cm 3 Use a calculator. Find the radius of the sphere. The circumference of a great circle is 2  r. So, 2  r = 30  or r = 15. Example 5: Find the volume of the sphere with a great circle that has a circumference of 30π centimeters. Round to the nearest tenth. Answer: The volume of the sphere is approximately 14,137.2 cm 3.

The volume of a hemisphere is one-half the volume of the sphere. Answer: The volume of the hemisphere is approximately 56.5 cubic feet. Example 6: Find the volume of the hemisphere with a diameter of 6 ft. Round to the nearest tenth. Volume of a hemisphere r = 3 ≈ 56.5 cm 3 Use a calculator.

Volume of a sphere r = 15 ≈ cm 3 Use a calculator. Example 7: Find the volume of the sphere. Round to the nearest hundredth and label the units. Answer: The volume of the sphere is approximately cm 3.

You know that the volume of the stone is 36,000  cubic inches. First use the volume formula to find the radius. Then find the diameter. Example 8: The stone spheres of Costa Rica were made by forming granodiorite boulders into spheres. One of the stone spheres has a volume of about 36,000 cubic inches. What is the diameter of the stone sphere? Volume of a sphere Replace V with 36,000  27,000 = r 3 Divide each side by The radius of the stone is 30 inches. So, the diameter is 2(30) or 60 inches. 30 = r Take the cube root of each side

You know that the volume of the jungle gym is 4,000  cubic feet. First use the volume formula to find the radius. Then find the diameter. Example 9: The jungle gym outside of Jada’s school is a perfect hemisphere. It has a volume of 4,000  cubic feet. What is the diameter of the jungle gym? Volume of a hemisphere Replace V with 4,000  6,000 = r 3 Divide each side by The radius of the jungle gym is 18.2 feet. So, the diameter is 2(18.2) or 36.4 feet ≈ r Take the cube root of each side

A.10.7 feet B.12.6 feet C.14.4 feet D.36.3 feet RECESS The jungle gym outside of Jada’s school is a perfect hemisphere. It has a volume of 4,000  cubic feet. What is the diameter of the jungle gym?