1. Lesson 10.6
Spheres pp

2. Objectives: 1. To define terms related to a sphere:
tangent plane, secant plane, and great circles.
2. To state and prove theorems about spheres.
3. To derive and apply a formula for arc length on a sphere.

3. Recall that a sphere is the set of points in space that are a given distance from a given point.

4. O is the center, OA is the radius, BC is a chord, DA is a diameter, EF is a tangent, BC is a secant.

5. Definition
A secant plane to a sphere is a plane that intersects a sphere in more than one point. n O

6. O n

7. Definition
A tangent plane to a sphere is a plane that intersects a sphere in exactly one point. The point is called the point of tangency.

8. S K R m

9. Theorem 10.19
The intersection of a sphere and a secant plane is a circle.

10. S C n B A D

11. Definition
A great circle of a sphere is the intersection of the sphere and a secant plane that contains the center of the sphere.

12. S m O

13. Theorem 10.20
Two points on a sphere that are not on the same diameter lie on exactly one great circle of the sphere.

14. O

15. Theorem 10.21
Two great circles of a sphere intersect at two points that are endpoints of a diameter of the sphere.

16. O

17. Theorem 10.22
All great circles of a sphere are congruent.

18. O

19. O

20. Theorem 10.23
A secant plane of a sphere is perpendicular to the line containing the center of the circle of intersection and the center of the sphere.

21. Theorem 10.24
A plane is tangent to a sphere if and only if it is perpendicular to the radius at the point of tangency.

22. O A B
To find the distance between A and B, you must find the length of the arc of the great circle that contains A and B.

23. A O B ) r 2 (
360
mAB
dAB p =

24. Homework pp

25. 5. mXAY = 35 A X Y
(2p•4) ≈ 2.4 units
360 35
dXY =

26. 9. mHJI = 160;
IJ = 3 J H I
(2p•3) ≈ 8.4 units
360
160
dHI =

27. Consider the earth to be a sphere
Consider the earth to be a sphere. If its diameter is 7900 miles, find the distance between two cities that are the given degree measures apart.
13. 36°
(2p•3950) ≈ mi.
360 36
d =

28. ■ Cumulative Review
Find each area.
21. 2 7

29. ■ Cumulative Review
Find each area.
22. 4 9

30. ■ Cumulative Review
Find each area.
23. 2 6 5

31. ■ Cumulative Review
Find each area.
24. 8

32. ■ Cumulative Review
Find each area.
25. 3