Created by:Christopher Hampton & James Nguyen

12.7 Surface Area of Spheres

Objectives

Recognize and know basic properties of spheres

Find surface area of spheres and hemispheres

Properties of Spheres

To visualize a sphere, consider infinitely many congruent circles in space, all with the same

point for their center. Considered together, these circles form a sphere. In space, a sphere is the locus of all points that are a given distance from

a given point called its center.

Properties of Spheres

A radius of a sphere is a segment with endpoints that are the center and a point on the sphere.

A chord is a segment with endpoints on the sphere itself.

A diameter is a chord that contains the center.

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

Vocabulary

Great circle- The intersection of a plane

and a sphere so that the plane contains

the center of the sphere. A great circle

has the same center as the sphere, and its radii are also radii of the sphere.

Hemisphere- Each great circle separates

a sphere into two congruent halves, each

being the hemisphere. Hemispheres have

a great circle as a base.

Surface Area of Spheres

If a sphere has a surface area of T square units and a radius of r units, then T=4pr2.

Remember a hemisphere is half a sphere plus its great circle base.

Example #1

In the figure, O is the center of the sphere, and plane R intersects the sphere in circle A. If AO = 3 centimeters and OB = 10 centimeters, find AB.

Example #1 Solution

Step 1: OB2 =AB2 + AO2 (Pythagorean Theorem)

Step 2: AB2+32 (OB=10, AO=3)

Step 3: 100 = AB2+9 (Simplify)

Step 4: 91=AB2 (Subtract 9 from each side).

Step 5: 9.5 ≈ AB (Square Root)

Try this: It's your turn Ali S.!

P is the center of the sphere

If Z is a point on circle T and PR = 11.6, find PZ.

Answer: 11.6 units

T

R

P

Example # 2

Find the surface area of a hemisphere.

C = 10p.

Example #2 Solution

Step 1: r=5 ( Formula for Circumference)

Step 2: 4p(5)2 (Surface Area of Spheres Formula)

Step 3: T≈ 314.2 units squared (Multiply)

Step 4: 314.2 / 2 = 157.1 units squared (Definition of a hemisphere)

Step 5: (5)p 2 =78.5 units squared

Step 6: 78.5 + 157.1 = 235.6 units squared (Solution)

Try this: It's your turn, Jimmy!

Find the surface of the sphere. Round to the nearest tenth.

Sphere: The area of a great circle is 512.7 square centimeters.

Answer: 2050.8 square centimeters

Example #3

Determine whether each statement is true or false. If false, give a counterexample.

The radii of a sphere are congruent to the radius of its great circle.

Example #3 Solution

Answer: True (The directions don't ask for an explanation.)

Try This : It's your turn!

Determine whether each statement is true or false. If false, give a counterexample.

Two spheres can intersect in one point.

Answer: True

Assignment:

Pg. 674 # 10-16

Just Kidding!!!

Horrible Homework!!! :(

Pg. 674 #3,4,9,10-24,26,28,37,38,41

Clay, good luck! :P

Dumb Math Jokes

1) Why is math the saddest subject? A: Because it has so many problems.

2) What do you get when you do math? A: Mostly Awful, Terrible Headaches

3) How can a mathematician beat 100 enemies? A: Multiply them by 0.

4) Mathematicians never die – they only lose some of their functions!

5) There are three kinds of people in the world; those who can count and those who can’t.