1. Created by: Christopher Hampton & James Nguyen

2. Objectives Recognize and know basic properties of spheres Find surface area of spheres and hemispheres

3. 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.

4. 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.

5. 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.

6. Surface Area of Spheres If a sphere has a surface area of T square units and a radius of r units, then T=4  r 2. Remember a hemisphere is half a sphere plus its great circle base.

7. 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.

8. Example #1 Solution Step 1: OB 2 =AB 2 + AO 2 (Pythagorean Theorem) Step 2: AB 2 +3 2 (OB=10, AO=3) Step 3: 100 = AB 2 +9 (Simplify) Step 4: 91=AB 2 (Subtract 9 from each side). Step 5: 9.5 ≈ AB (Square Root)

9. 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

10. Example # 2 Find the surface area of a hemisphere. C = 10 .

11. Example #2 Solution Step 1: r=5 ( Formula for Circumference) Step 2: 4  (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:  2 =78.5 units squared Step 6: 78.5 + 157.1 = 235.6 units squared (Solution)

12. 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

13. 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.

14. Example #3 Solution Answer: True (The directions don't ask for an explanation.)

15. 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

16. Assignment: Pg. 674 # 10-16 Just Kidding!!!

17. Horrible Homework!!! :( Pg. 674 #3,4,9,10-24,26,28,37,38,41 Clay, good luck! :P

18. 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.