Holt Geometry

10-8 Spheres

Learn and apply the formula for the volume of a sphere.Learn and apply the formula for the surface area of a sphere.

Objectives

Holt Geometry

10-8 Spheres

spherecenter of a sphereradius of a spherehemispheregreat circle

Vocabulary

Holt Geometry

10-8 Spheres

A sphere is the locus of points in space that are a fixed distance from a given point called the center of a sphere. A radius of a sphere connects the center of the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres

Holt Geometry

10-8 Spheres

Holt Geometry

10-8 Spheres

Example 1A: Finding Volumes of Spheres

Find the volume of the sphere. Give your answer in terms of .

= 2304 in3 Simplify.

Volume of a sphere.

Holt Geometry

10-8 Spheres

Example 1B: Finding Volumes of Spheres

Find the diameter of a sphere with volume 36,000 cm3.

Substitute 36,000 for V.

27,000 = r3

r = 30

d = 60 cm d = 2r

Take the cube root of both sides.

Volume of a sphere.

Holt Geometry

10-8 Spheres

Example 1C: Finding Volumes of Spheres

Find the volume of the hemisphere.

Volume of a hemisphere

Substitute 15 for r.

= 2250 m3 Simplify.

Holt Geometry

10-8 Spheres

Check It Out! Example 1

Find the radius of a sphere with volume 2304 ft3.

Volume of a sphere

Substitute for V.

r = 12 ft Simplify.

Holt Geometry

10-8 Spheres

Example 2: Sports Application

A sporting goods store sells exercise balls in two sizes, standard (22-in. diameter) and jumbo (34-in. diameter). How many times as great is the volume of a jumbo ball as the volume of a standard ball?

standard ball: jumbo ball:

A jumbo ball is about 3.7 times as great in volume as a standard ball.

Holt Geometry

10-8 Spheres

Holt Geometry

10-8 Spheres

Example 3A: Finding Surface Area of Spheres

Find the surface area of a sphere with diameter 76 cm. Give your answers in terms of .

S = 4r2

S = 4(38)2 = 5776 cm2

Surface area of a sphere

Holt Geometry

10-8 Spheres

Example 3B: Finding Surface Area of Spheres

Find the volume of a sphere with surface area 324 in2. Give your answers in terms of .

Substitute 324 for S.324 = 4r2

r = 9 Solve for r.

Substitute 9 for r.

The volume of the sphere is 972 in2.

S = 4r2 Surface area of a sphere

Holt Geometry

10-8 Spheres

Example 3C: Finding Surface Area of Spheres

Find the surface area of a sphere with a great circle that has an area of 49 mi2.

Substitute 49 for A.49 = r2

r = 7 Solve for r.

S = 4r2

= 4(7)2 = 196 mi2 Substitute 7 for r.

A = r2 Area of a circle

Holt Geometry

10-8 Spheres

Check It Out! Example 3

Find the surface area of the sphere.

Substitute 25 for r.

S = 2500 cm2

S = 4r2

S = 4(25)2

Surface area of a sphere

Holt Geometry

10-8 Spheres

Example 4: Exploring Effects of Changing Dimensions

The radius of the sphere is

multiplied by . Describe the

effect on the volume.

original dimensions: radius multiplied by :

Notice that . If the radius is multiplied by

, the volume is multiplied by , or .

Holt Geometry

10-8 Spheres

Check It Out! Example 4

The radius of the sphere is divided by 3. Describe the effect on the surface area.

original dimensions: dimensions divided by 3:

The surface area is divided by 9.

S = 4r2

= 4(3)2 = 36 m3

S = 4r2

= 4(1)2 = 4 m3

Holt Geometry

10-8 Spheres

Example 5: Finding Surface Areas and Volumes of Composite Figures

Find the surface area and volume of the composite figure. Give your answer in terms of .

Step 1 Find the surface area of the composite figure.

The surface area of the composite figure is the sum of the curved surface area of the hemisphere, the lateral area of the cylinder, and the base area of the cylinder.

Holt Geometry

10-8 Spheres

Example 5 Continued

The surface area of the composite figure is

L(cylinder) = 2rh = 2(6)(9) = 108 in2

B(cylinder) = r2 = (6)2 = 36 in2

72 + 108 + 36 = 216 in2.

Find the surface area and volume of the composite figure. Give your answer in terms of .

Holt Geometry

10-8 Spheres

Step 2 Find the volume of the composite figure.

Example 5 Continued

Find the surface area and volume of the composite figure. Give your answer in terms of .

The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the cylinder.

The volume of the composite figure is 144 + 324 = 468 in3.

Holt Geometry

10-8 Spheres

Check It Out! Example 5

Find the surface area and volume of the composite figure.

Step 1 Find the surface area of the composite figure.

The surface area of the composite figure is the sum of the curved surface area of the hemisphere, the lateral area of the cylinder, and the base area of the cylinder.

Holt Geometry

10-8 Spheres

Check It Out! Example 5 Continued

The surface area of the composite figure is

Find the surface area and volume of the composite figure.

L(cylinder) = 2rh = 2(3)(5) = 30 ft2

B(cylinder) = r2 = (3)2 = 9 ft2

18 + 30 + 9 = 57 ft2.

Holt Geometry

10-8 Spheres

Step 2 Find the volume of the composite figure.

Find the surface area and volume of the composite figure.

Check It Out! Example 5 Continued

The volume of the composite figure is the volume of the cylinder minus the volume of the hemisphere.

V = 45 – 18 = 27 ft3