Section 12.6Surface Areas and Volumes of Spheres

S = 4r2 Surface area of a sphere

= 4(4.5)2 Replace r with 4.5.

≈ 254.5 Simplify.

Answer: 254.5 in2

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 41.07 mm2

Replace r with 3.7.

Use a calculator.= 41.07

2 214

2S r r

2 214 3.7 3.7

2S

First, find the radius. The circumference of a great circle is 2r. So, 2r = 10 or r = 5.

Example 3: Find the surface area of the sphere if the circumference of the great circle is 10π.

S = 4r2 Surface area of a sphere

= 4(5)2 Replace r with 5.

= 100 Use a calculator.

Answer: 100 ft2

First, find the radius. The area of a great circle is r2. So, r2 = 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 886.7 m2

S = 4r2 Surface area of a sphere

≈ 4(8.4)2 Replace r with 5.

≈ 886.7 Use a calculator.

Volume of a sphere

r = 15

≈ 14,137.2 cm3 Use a calculator.

Find the radius of the sphere. The circumference of a great circle is 2r. So, 2r = 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.

34

3V r

3415

3V

Answer: The volume of the sphere is approximately 14,137.2 cm3.

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 cm3 Use a calculator.

31 4

2 3V r

323

3V

Volume of a sphere

r = 15

≈ 2144.7 cm3 Use a calculator.

Example 7: Find the volume of the sphere. Round to the nearest hundredth and label the units.

34

3V r

348

3V

Answer: The volume of the sphere is approximately 2144.7 cm3.

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 = r3 Divide each side by

34

3V r

3436,000

3r

4

3

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 = r3 Divide each side by

31 4

2 3V r

32

4,0003

r

2

3

The radius of the jungle gym is 18.2 feet. So, the diameter is 2(18.2) or 36.4 feet.

18.2 ≈ 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?