1. Trigonometry and Regular Polygons
Lesson 14.6
Trigonometry and Regular Polygons pp

2. Objectives:
1. To identify and apply the relation between the apothem and side or a regular polygon.
2. To apply trigonometry to regular polygons.

3. EXAMPLE 1 Find the apothem and radius of a regular pentagon with a side length of 12.
72°

4. EXAMPLE 1 Find the apothem and radius of a regular pentagon with a side length of 12.

5. EXAMPLE 1 Find the apothem and radius of a regular pentagon with a side length of 12.
36° a 6

6. EXAMPLE 1 Find the apothem and radius of a regular pentagon with a side length of 12.
36° 6

7. 1. Find the measure of the central angle.
Practice: Use a regular octagon of side 10. Round segment lengths to the nearest tenth and angle measures to the nearest degree.
1. Find the measure of the central angle.
45°

8. Practice: Use a regular octagon of side 10
Practice: Use a regular octagon of side 10. Round segment lengths to the nearest tenth and angle measures to the nearest degree.
2. Find the apothem.

9. Practice: Use a regular octagon of side 10
Practice: Use a regular octagon of side 10. Round segment lengths to the nearest tenth and angle measures to the nearest degree.
2. Find the apothem.

10. Practice: Use a regular octagon of side 10
Practice: Use a regular octagon of side 10. Round segment lengths to the nearest tenth and angle measures to the nearest degree.
2. Find the apothem.
22.5° a 5

11. Practice: Use a regular octagon of side 10
Practice: Use a regular octagon of side 10. Round segment lengths to the nearest tenth and angle measures to the nearest degree.
3. Find the radius. r
22.5° 5

12. EXAMPLE 2 If the radius of a circle is 10 cm, find the length of the side of an inscribed regular pentagon. 10

13. Homework pp

14. ►A. Exercises 1. mCPD and mCPM
Refer to the regular pentagon to find the following.
1. mCPD and mCPM C A 6 B D M E P

15. ►A. Exercises
Refer to the regular pentagon to find the following.
2. mPCM C A 6 B D M E P

16. ►A. Exercises
Refer to the regular pentagon to find the following.
3. PM and PD C A 6 B D M E P

17. ►A. Exercises 4. area of the pentagon
Refer to the regular pentagon to find the following.
4. area of the pentagon C A 6 B D M E P

18. 5. the measure of the central angle and its bisected angle.
►B. Exercises
Using a regular decagon of side s, find
5. the measure of the central angle and its bisected angle. s

19. 6. the apothem in terms of the side.
►B. Exercises
Using a regular decagon of side s, find
6. the apothem in terms of the side. s

20. 7. the radius in terms of the side.
►B. Exercises
Using a regular decagon of side s, find
7. the radius in terms of the side. s

21. 8. the apothem and radius if the side is 10 mm.
►B. Exercises
Using a regular decagon of side s, find
8. the apothem and radius if the side is 10 mm. 10

22. 9. the side if the circumscribed circle has a radius of 12 m.
►B. Exercises
Using a regular decagon of side s, find
9. the side if the circumscribed circle has a radius of 12 m. 12

23. 10. the area if the side is 7 feet.
►B. Exercises
Using a regular decagon of side s, find
10. the area if the side is 7 feet. 7

24. ►B. Exercises 11. apothem and radius.
Use a regular octagon of side s. Find formulas for the
11. apothem and radius. s

25. ►B. Exercises
Use a regular octagon of side s. Find formulas for the
12. area. s

26. 13. If the apothem is 7 in., how long is the side?
►B. Exercises
Use a regular octagon of side s.
13. If the apothem is 7 in., how long is the side?

27. 25. Cube with a volume of 64 cu. in.
■ Cumulative Review
Find the dimensions of each.
25. Cube with a volume of 64 cu. in.

28. 26. Sphere with a volume of 36 cu. ft.
■ Cumulative Review
Find the dimensions of each.
26. Sphere with a volume of  cu. ft.

29. 27. Regular octahedron with a volume of 72 2 cubic inches.
■ Cumulative Review
Find the dimensions of each.
27. Regular octahedron with a volume of cubic inches.

30. ■ Cumulative Review
Find the volume of each.
28. A rectangular solid has a square base with edges 5 cm long. The lateral faces are golden rectangles.

31. ■ Cumulative Review
Find the volume of each.
29. A pyramid is 11 units high and has a regular polygon as its base. The polygon has 20 sides each 4 units long. (Hint: Use trigonometry to find the apothem first.)