1. 1 Geometry Section 6-3A Regular Polygons Page 442

2. 2 Regular Polygons: Regular Polygons - Equilateral and equiangular. Try It: Equilateral, Equiangular or Regular? Equilateral Regular Equiangular Pg. 442

3. 3 Regular Polygons: Regular Polygons Vocabulary Center of a regular polygon: point of intersection of the perpendicular bisectors of the sides. Radius: segment from a vertex to the center. All of the radii of a regular polygon are congruent. Apothem: perpendicular segment from the center to the side. Pg. 442

4. 4 Regular Polygons: Any regular polygon can be divided into congruent isosceles triangles by drawing all of the radii. Pg. 443

5. 5 Explore: (fill in the chart) How many sides does the polygon have? Pg. 443 3 Into how many triangles is it divided by the radii? 3 What is the area of each of these triangles in terms of a and s? as 1212 What is the area of the polygon in terms of a and s? as 3232 s Note: the length of the apothem is the height of the triangle.

6. 6 Explore: (fill in the chart) How many sides does the polygon have? Pg. 443 4 Into how many triangles is it divided by the radii? 4 What is the area of each of these triangles in terms of a and s? as 1212 What is the area of the polygon in terms of a and s? 2as

7. 7 Explore: (fill in the chart) How many sides does the polygon have? Pg. 443 6 Into how many triangles is it divided by the radii? 6 What is the area of each of these triangles in terms of a and s? as 1212 What is the area of each of the polygon in terms of a and s? 3as

8. 8 Explore: (fill in the chart) Pg. 443 as 1212 2as # of sides (n)# of trianglesArea of triangleArea of polygon 33 44 66 as 1212 1212 3232 3as Write a formula for the area of a regular n-gon in terms of n, a, and s. If a regular n-gon has apothem a and perimeter p, write a formula for its area in terms of a and p. ap 1212 nas 1212 Because n(s) = perimeter

9. 9 Example: Pg. 444 Find the area of the regular pentagon. Since a regular pentagon has 5 congruent sides, the perimeter is 5 x 10 = 50. Area = ap 1212 =.5(6.9)(50) = 172.5 in 2

10. 10 Example: Pg. 444 Find the area of the regular octagon. Since a regular octagon has 8 congruent sides, the perimeter is 8 x 15 = 120. Area = ap 1212 =.5(18.1)(120) = 1086 cm 2

11. 11 Theorems: Pg. 444

12. 12 Exercises: #4 Pg. 445 Find the measure of one interior angle and one exterior angle of a regular heptagon. (7-2)180 7 (n-2)180 n  128.57 o 360 n 360 7  51.43 o

13. 13 Exercises: #5 Pg. 445 Find the measure of one interior angle and one exterior angle of a regular octagon. (8-2)180 8 (n-2)180 n  135 o 360 n 360 8  o

14. 14 Exercises: #6 Pg. 445 Find the measure of one interior angle and one exterior angle of a regular decagon. (10-2)180 10 (n-2)180 n  144 o 360 n 360 10  o

15. 15 Exercises: #8 Pg. 445 One angle of a regular polygon measures 160 o. How many sides does the polygon have? (n-2)180 n 180n – 360 = 160n  160 o Multiply both sides by n. 20n = 360 n = 18

16. 16 Exercises: #9 Pg. 445 Is it possible for a regular polygon to have an interior angle of 156 o ? Explain. (n-2)180 n 180n – 360 = 156n  156 o Multiply both sides by n. 24n = 360 n = 15 yes If you don’t end up with an exact number, then it is not possible.

17. 17 Exercises: #10 Pg. 445 As the number of sides of a regular polygon increases, what happens to the size of each interior angle of the polygon? Explain. The measures of the interior angles will increase because the angles have to “spread out” to make room for another side.

18. 18 Exercises: #11 - 13 Pg. 445 The perpendicular bisectors of the sides of a regular polygon meet at its. The distance from the center of a regular polygon to one of its sides is the of the polygon. center apothem The distance from the center of a regular polygon to one of its vertices is the of the polygon. radius

19. 19 Exercises: #15 Pg. 445 Find the area of the regular pentagon. Since a regular pentagon has 5 congruent sides, the perimeter is 5 x 30 = 150. Area = ap 1212 =.5(20.6)(150) = 1545 in 2

20. 20 Exercises: #16 Pg. 445 Find the area of the regular octagon. Since a regular octagon has 8 congruent sides, the perimeter is 8 x 2 = 16. Area = ap 1212 =.5(2.4)(16) = 19.2 in 2

21. 21 Exercises: #17 Pg. 445 Find the area of the square. Since a square has 4 congruent sides, the perimeter is 4 x 8 = 32. Area = ap 1212 =.5(4)(32) = 64 m 2 What is the length of 1 side?

22. 22 Exercises: #18 Pg. 445 Find the area of a regular decagon with apothem 6.8 m and side length 4.4 m. Since a regular decagon has 10 congruent sides, the perimeter is 10 x 4.4 = 44. Area = ap 1212 =.5(6.8)(44) = 149.6 m 2

23. 23 Exercises: #20 Pg. 446 Find the apothem and radius. Then find the area. Use the ratio for a 45-45-90 triangle. Area = ap 1212 =.5(6)(48) = 144 Look a the triangle. What type of “special” right triangle is this? a = 6, r = 6  2

24. 24 Exercises: #21 Pg. 446 Find the apothem and radius. Then find the area. Use the ratio for a 30-60-90 triangle. Area = ap 1212 =.5(10.39)(72)  374.04 Look a the triangle. What type of “special” right triangle is this? a = 6  3, r = 12

25. 25 Homework: Practice 6-3A Test Thursday