12.6 Surface Area and Volume of Spheres - RJS … · Surface Area and Volume of Spheres THEOREM 12.11: SURFACE AREA OF A SPHERE The surface area S of a sphere is r S 5 , where r …

Your Notes

Surface Area and Volume of Spheres

THEOREM 12.11: SURFACE AREA OF A SPHERE

The surface area S of a sphere is r

S 5 ,

where r is the radius of the sphere.

12.6Goal p Find surface areas and volumes of spheres.

VOCABULARY

Sphere

Center of a sphere

Radius of a sphere

Chord of a sphere

Diameter of a sphere

Tangent

Great circle

Hemisphere

Copyright © Holt McDougal. All rights reserved. Lesson 12.6 • Geometry Notetaking Guide 345

Your Notes

Surface Area and Volume of Spheres

THEOREM 12.11: SURFACE AREA OF A SPHERE

The surface area S of a sphere is r

S 5 4πr2 ,


12.6Goal p Find surface areas and volumes of spheres.

VOCABULARY

Sphere A sphere is the set of all points in space equidistant from a given point.

Center of a sphere The center of a sphere is the given point from which all points on the sphere are equidistant.

Radius of a sphere A radius of a sphere is a segment from the center to a point on the sphere.

Chord of a sphere A chord of a sphere is a segment whose endpoints are on the sphere.

Diameter of a sphere A diameter of a sphere is a chord that contains the center of the sphere.

Tangent A tangent is a line or plane that intersects the sphere in exactly one point.

Great circle A great circle is the intersection of a sphere and a plane that contains the center of the sphere.

Hemisphere A hemisphere is one of the congruent halves of a sphere.


LAH_GE_11_FL_NTG_Ch12_325-355.indd 345LAH_GE_11_FL_NTG_Ch12_325-355.indd 345 2/4/09 12:36:51 AM2/4/09 12:36:51 AM

Find the surface area of the sphere.

7 cm

SolutionS 5 4πr2 Formula for surface area of a sphere

5 4π( )2 Substitute for r.

5 π Simplify.

< Use a calculator.

The surface area of the sphere is about cm2.

Example 1 Find the surface area of a sphere

The surface area of a sphere is 110.25π square feet. Find the diameter of the sphere.

Solution S 5 4πr2 Formula for surface area of a sphere

5 4πr2 Substitute for S.

5 r2 Divide each side by .

5 r Find the positive square root.

The diameter of the sphere is

2r 5 2 p 5 feet.

Example 2 Find the diameter of a sphere

Be sure to multiply the value of r by 2 to find the diameter.

346 Lesson 12.6 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Your Notes


Find the surface area of the sphere.

7 cm

SolutionS 5 4πr2 Formula for surface area of a sphere

5 4π( 7 )2 Substitute 7 for r.

5 196 π Simplify.

< 615.75 Use a calculator.

The surface area of the sphere is about 615.75 cm2.

Example 1 Find the surface area of a sphere

The surface area of a sphere is 110.25π square feet. Find the diameter of the sphere.

Solution S 5 4πr2 Formula for surface area of a sphere

110.25π 5 4πr2 Substitute 110.25π for S.

27.5625 5 r2 Divide each side by 4π .

5.25 5 r Find the positive square root.

The diameter of the sphere is

2r 5 2 p 5.25 5 10.5 feet.

Example 2 Find the diameter of a sphere

Be sure to multiply the value of r by 2 to find the diameter.


Your Notes


THEOREM 12.12: VOLUME OF A SPHERE

The volume V of a sphere is r

V 5 ,


Find the volume of the sphere. 3 mm

Solution

V 5 4 } 3 πr3 Formula for volume of a sphere

5 4 } 3 π( )3 Substitute for r.

< Use a calculator.

The volume of the sphere is about cubic millimeters.

Example 3 Find the volume of a sphere

Find the volume of the composite solid.

5 in.

3 in.

SolutionVolume of solid

5 Volume of

2 Volume of

5 πr2h 2 1 } 2 1 4 } 3 πr3 2 Volume formulas

5 π( )2 ( ) 2 1 } 2 1 4 } 3 π( )3 2 Substitute.

5 π 2 π Simplify.

< Use a calculator.

The volume of the solid is about cubic inches.

Example 4 Find the volume of a composite solid


Your Notes


THEOREM 12.12: VOLUME OF A SPHERE

The volume V of a sphere is r

V 5 4 } 3 πr3 ,


Find the volume of the sphere. 3 mm

Solution

V 5 4 } 3 πr3 Formula for volume of a sphere

5 4 } 3 π( 3 )3 Substitute 3 for r.


The volume of the sphere is about 113.1 cubic millimeters.

Example 3 Find the volume of a sphere

Find the volume of the composite solid.

5 in.

3 in.

SolutionVolume of solid

5 Volume of cylinder

2 Volume of hemisphere

5 πr2h 2 1 } 2 1 4 } 3 πr3 2 Volume formulas

5 π( 3 )2 ( 5 ) 2 1 } 2 1 4 } 3 π( 3 )3 2 Substitute.

5 45 π 2 18 π Simplify.


The volume of the solid is about 84.82 cubic inches.

Example 4 Find the volume of a composite solid


Your Notes


1. The diameter of a sphere is 1 }

Ï}

π meter. Find the

surface area of the sphere.

2. The surface area of a sphere is 169π square inches. Find the radius of the sphere.

3. The radius of a sphere is 2.4 cm. Find the volume of the sphere. Round your answer to two decimal places.

4. Find the volume of the composite

5.8 ft

14 ft

solid. Round your answer to two decimal places.

Checkpoint Complete the following exercises.


Homework

Your Notes


1. The diameter of a sphere is 1 }

Ï}

π meter. Find the

surface area of the sphere.

1 m2

2. The surface area of a sphere is 169π square inches. Find the radius of the sphere.

6.5 in.

3. The radius of a sphere is 2.4 cm. Find the volume of the sphere. Round your answer to two decimal places.

57.91 cm3

4. Find the volume of the composite

5.8 ft

14 ft

solid. Round your answer to two decimal places.

1888.21 ft3

Checkpoint Complete the following exercises.


Homework

Your Notes


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12.6 Surface Area and Volume of Spheres - RJS … · Surface Area and Volume of Spheres THEOREM 12.11: SURFACE AREA OF A SPHERE The surface area S of a sphere is r S 5 , where r …