section 12.6 surface areas and volumes of spheres
TRANSCRIPT
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Section 12.6Surface Areas and Volumes of Spheres
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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.
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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.
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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
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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
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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.
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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.
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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
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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.
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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
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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
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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?