Chapter 12.7

Surface Areas of Spheres

Objectives

Recognize and define basic properties of spheres

Find surface areas of spheres

• Point D is the center of the sphere

• AB is the diameter of sphere D

• DC, DA, and DB and radii

• FG and AB are chords

• JH is a tangent to sphere D at point E

Parts of a Sphere

- The intersection of a plane and a sphere can be or .

- When a plane intersects a sphere so that it contains the center of the sphere, the intersection is called .

(Note: A great circle has the same center as the sphere, and its radii are also radii of the sphere.)

No Intersection

A PointA Circle

A great Circle

Center

Each Great circle

divides a

sphere into two halves, each

called a hemi-

sphere

.

In the figure, C is the center of the sphere, and plane R intersects the sphere in circle X. If XC = 9 centimeters and CY = 30 centimeters, find XY. Triangle CXY is a right triangle. (Angle X = 90°)

R 30 cm

9 cmXY² + XC² = YC² Pythagorean Theorem

XY² + 9² = 30² Plug in numbers

XY² + 81 = 900 Square Numbers

XY² = 900 – 81 Subtract 81 from both sides

XY² = 819 900 – 81 = 819

XY = √819 Find the square root of 819

XY ≈ 28.6 cm Punch it in the calculator…and

you get the approximate answer

Example 1:

If a sphere has a

surface area of A

square units and a

radius of r units,

then A = 4πr².

Area of a Sphere

(A great circle’s area is πr²)

Example 2:

Find the surface area of the sphere given the area of the great circle.

We know that the surface area of a sphere is four times the area of the great circle.

A = 4πr² Surface Area of a sphere

≈ 4(603.3) πr² ≈ 603.3

≈ 2413.2 Multiply

The surface area of this

sphere is ≈ 2413.2 in.²

G ≈ 603.3 in.²

Find the surface area of the hemisphere.

A hemisphere is half of a sphere. To find the surface area, find half of the surface area of the sphere and add the area of the great circle.

8.4 cm

Surface area = ½ (4πr²) + πr²

Surface area of a hemisphere

= ½ [4π(8.4)²] + π(8.4)²

Substitution

≈ 664.7

Use a calculator

The surface area of the hemisphere is approximately 664.7cm²

Example 3:

Find the surface area of a baseball given the circumference of 9 inches to determine how much leather is needed to cover the ball.

First find the radius of the ball.

C = 2πr Circumference of a circle

9 = 2πr Substitution

9/2π = r Division

1.4 ≈ r Use a calculator

Next, find the surface area.

A = 4πr² Surface area of a sphere

≈ 4π(1.4)² Substitution

≈ 25.8 Use a calculator

The surface area of the baseball is approximately 25.8 inches²

Assignment

Pre-AP GeometryPage 674

# 10-29, #34 and #36