find the volume of each

Find the volume of each.

V = (Bh)/3 = (5×9)(12)/3 = 180 ft3

V = (Bh)/3 650 = (130)h/3 = 15 m

V = (Bh)/3 = π(42)(8)/3 = 42.7π mm3

V = (Bh)/3 = π(√(4.22)(14.6)/3 = 85.8π m3

Find the volume of a cone with a diameter of 8.4 meters and a height of 14.6 meters.

Find the height of a hexagonal pyramid with a base area of 130 square meters and a volume of 650 cubic meters.

Ch 12.6Volumes & Surface Areas

of Spheres

Standard 9.0Students compute the volume and surface area

of spheres.

Learning Target:I will be able to solve problems involving the volume and surface areas of spheres.

great circleA circle formed when a planeintersects with its center at thecenter of the sphere.

poleThe endpoints of the diameter of a great circle.

hemisphereOne of two congruent parts into which a great circle separates a sphere.

great circle

Theorem 12-13

Surface Area of a Sphere

Find the surface area of the sphere in terms of π.

S = 4r2 Surface area of a sphere

= 4(4.5)2 r = 4.5.

≈ 81π Multiply.

Answer: about 254 .6 in2

A. 147.2π in2

B. 150.5π in2

C. 153.6π in2

D. 156.3π in2

Find the surface area of the sphere.

S = 4r2 Surface area of a sphere

= 4(6.25)2 r = 4.5.

=156.3π Multiply.

Use Great Circles to Find Surface Area

A. Find the surface area of the hemisphere.

Find half the area of a sphere with the radius of 3.7 millimeters. Then add the area of the great circle.

Surface area of a hemisphere

r = 3.7.

Multiply.= 41π Answer: about 129.0 mm2

Use Great Circles to Find Surface Area

B. Find the surface area of a sphere if the circumference of the great circle is 10 feet.

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

S = 4r2 Surface area of a sphere

= 4(5)2 r = 5.

= 100π Multiply.

Answer: about 314.2 ft2

Use Great Circles to Find Surface Area

C. Find the surface area of a sphere if the area of the great circle is approximately 220 m2.

First, find the radius. The area of a great circle is r2. So, r2 = 220 or r =√220/π.

S = 4r2 Surface area of a sphere

= 4(√220/π)2 r = √(220/π)

= 880 Multiply.

Answer: 880 m2

A. 35.3π m2

B. 52.9π m2

C. 53.9π m2

D. 55.0π m2

A. Find the surface area of the hemisphere.

r = 4.2

Multiply.= 52.92π

Surface area of a hemisphere

(4.2)2 (4.2)2

A. 32π ft2

B. 64π ft2

C. 128π ft2

D. 256π ft2

B. Find the surface area of a sphere if the circumference of the great circle is 8 feet.

S = 4r2 Surface area of a sphere

= 4(4)2 r = 8π/2π.

= 64π Multiply.

A. 320 ft2

B. 440 ft2

C. 640 ft2

D. 720 ft2

C. Find the surface area of the sphere if the area of the great circle is approximately 160 square meters.

S = 4r2 Surface area of a sphere

= 4(√160/π)2 r = √(160/π)

= 640 ft2 Multiply.

Theorem 12-14

Volumes of Spheres and Hemispheres

A. Find the volume a sphere with a great circle circumference of 30 centimeters in terms of π.

Volume of a sphere

(15)3

r = 15= 4500π cm3

Multiply.

Find the radius of the sphere. The circumference of a great circle is 2r. So, 2r = 30 or r = 15.

Answer: about 14,137.2 cm3.

Volumes of Spheres and Hemispheres

B. Find the volume of the hemisphere with a diameter of 6 feet. Round to the nearest tenth.

The volume of a hemisphere is one-half the volume of the sphere.

Answer: about 56.5 cubic feet.

Volume of a hemisphere

r 3

= 18π ft3

Multiply.

A. 85.3π cm3

B. 511.8π cm3

C. 682.7π cm3

D. 2047.2π cm3

A. Find the volume of the sphere.

Volume of a sphere

(8)3

r = 8= 682.7π cm3

Multiply.

A. 1066π m3

B. 2132π m3

C. 8554π m3

D. 42667π m3

B. Find the volume of the hemisphere.

Volume of a hemisphere

r 3

= 42667π m3

Multiply.

(40)3

Solve Problems Involving Solids

ARCHEOLOGY 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?

Understand You know that the volume of the stoneis 36,000 cubic inches.

Plan First use the volume formula to find theradius. Then find the diameter.

Solve Problems Involving Solids

Replace V with 36,000.

Solve Volume of a sphere

Divide each side by

Use a calculator to find

÷ ENTER( )2700 1 3 30

The radius of the stone is 30 inches. So, the diameter is 2(30) or 60 inches.

Answer: 60 inches

Solve Problems Involving Solids

CHECK You can work backward to check thesolution.

If the diameter is 60, then r = 30. If r = 30,

then V = cubic

inches.

The solution is correct.

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?

Volume of a hemisphere

V = 4000π, r = d/2

= 36.3 ft3 Multiply.

4000π = π 3 23

d2