Holt Geometry

10-8 Spheres10-8 Spheres

Holt Geometry

Holt Geometry

10-8 Spheres

Warm UpFind each measurement.

1. the radius of circle M if the diameter is 25 cm

2. the circumference of circle X if the radius is

42.5 in.

3. the area of circle T if the diameter is 26 ft

4. the circumference of circle N if the area is

625 cm2

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

The figure shows a hemisphere and a cylinder with a cone removed from its interior. The cross sections have the same area at every level, so the volumes are equal by Cavalieri’s Principle. You will prove that the cross sections have equal areas in Exercise 39.

The height of the hemisphere is equal to the radius.

Holt Geometry

10-8 Spheres

V(hemisphere) = V(cylinder) – V(cone)

The volume of a sphere with radius r is twice the volume of the hemisphere, or .

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.

Holt Geometry

10-8 Spheres

Example 1C: Finding Volumes of Spheres

Find the volume of the hemisphere.

Holt Geometry

10-8 Spheres

In the figure, the vertex of the pyramid is at the center of the sphere. The height of the pyramid is approximately the radius r of the sphere. Suppose the entire sphere is filled with n pyramids that each have base area B and height r.

Holt Geometry

10-8 Spheres

4r2 ≈ nB

If the pyramids fill the sphere, the total area of the bases is approximately equal to the surface area of the sphere S, so 4r2 ≈ S. As the number of pyramids increases, the approximation gets closer to the actual surface area.

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 .

Holt Geometry

10-8 Spheres

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.

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.

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

Lesson Quiz: Part I

Find each measurement. Give your answers in terms of .

1. the volume and surface areaof the sphere

2. the volume and surface area of a sphere with great circle area 36 in2

3. the volume and surface areaof the hemisphere

Holt Geometry

10-8 Spheres

Lesson Quiz: Part II

4. A sphere has radius 4. If the radius is multiplied

by 5, describe what happens to the surface area.

5. Find the volume and surface area of the

composite figure. Give your answer in terms of .