Chapter 12.7 Surface Areas of Spheres
Jan 19, 2016
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