12-6 Surface Area and Volume of Spheres

•You found surface areas of prisms and cylinders.

• Find surface areas of spheres.

• Find volumes of spheres.

How many bases…..

• Does a prism have?

• Does a pyramid have?

• Does a cylinder have?

• Does a cone have?

• Does a sphere have?

2

1

2

1

0

Sphere Measuring

What one measurement do you need to know when working with a sphere?

The center?

The diameter?

The radius?

No

?

yes

The surface area of a sphere is 4π times the square of its radius.

SA = 4 π r2

P. 880

Find the Surface Area of the Sphere

SA = 4π r2

SA = 4π(5)2

SA = 100π (exact)

SA ≈ 314.2 (approx)

r = 5

•Find the surface area of the sphere. Round to the nearest tenth.

• S = 4r2 Surface area of a sphere

• = 4(4.5)2 Replace r with 4.5.

• ≈ 254.5 Simplify.

• Answer: 254.5 in2

• A. 462.7 in2

• B. 473.1 in2

• C. 482.6 in2

• D. 490.9 in2

•Find the surface area of the sphere. Round to the nearest tenth.

•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

• Answer: about 129.0 mm2

•Replace r with 3.7.

•Use a calculator.•≈ 129.0

•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.

Answer: about 314.2 ft2

• S = 4r2 Surface area of a sphere

• = 4(5)2 Replace r with 5.

• ≈ 314.2 Use a calculator.

•C. Find the surface area of a sphere if the area of the great circle is approximately 220 square meters.

•First, find the radius. The area of a great circle is r2. So, r2 = 220 or r ≈ 8.4.

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.

• A. 110.8 m2

• B. 166.3 m2

• C. 169.5 m2

• D. 172.8 m2

•A. Find the surface area of the hemisphere.

• A. 100.5 ft2

• B. 201.1 ft2

• C. 402.2 ft2

• D. 804.3 ft2

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

• 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.

The volume of a sphere is four-thirds the product of π and the cube of its radius.

V = 4/3 π r 3

P. 882

Find the Volume of the Sphere

V = 4/3 π r 3

V = 4/3 π (5) 3

V = 166⅔π (exact)

V ≈ 523.6 (approx)

r = 5

•A. Find the volume a sphere with a great circle circumference of 30 centimeters. Round to the nearest tenth.

•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.

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

• A. 268.1 cm3

• B. 1608.5 cm3

• C. 2144.7 cm3

• D. 6434 cm3

•A. Find the volume of the sphere to the nearest tenth.

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: The volume of the hemisphere is approximately 56.5 cubic feet.

•Volume of a hemisphere

•Use a calculator.

•r =3

• A. 3351.0 m3

• B. 6702.1 m3

• C. 268,082.6 m3

• D. 134,041.3 m3

•B. Find the volume of the hemisphere to the nearest tenth.

Find the Volume and the Surface Area of a sphere…

When the area of one its great circles is 4π.

What do circle area and sphere volume/surface area have in common?

Radius!!!

A = 4π

Find the Volume and the Surface Area of a sphere…A = πr2

4π = πr2

r2 = 4

r = 2

V = 4/3 π r 3

V = 4/3 π (2) 3

V= 10⅔π

V ≈ 33.5

A = 4π

SA = 4πr2

SA = 4π(2)2

SA = 16π

SA ≈ 50.3

•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.

• 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

CHECK You can work backward to check the solution.

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?

Find the volume of a sphere whose surface area is 36π

SA = 4 π r2

36π = 4π r2

9 = r2

r = 3

V = 4/3 π r 3

V = 4/3 π (3)3

V = 36π

V ≈ 113.1

SA =36π

12-6 Assignment

Page 884, 10-15, 18-22

No Formulas with numbers

No Credit