1. Lesson 11-4 Areas of Regular Polygons (page 440) Essential Question How can you calculate the area of any figure?

2. Areas of Regular Polygons

3. Student Activity Using a compass, construct a circle with a given radius.

4. r

5. Mark off congruent arcs with chords equal to the radius of the circle. Start where the radius meets the circle.

6. r

7. Connect the consecutive points on the circle to form a polygon.

8. r r r r r r r

9. r r r r r r r The figure formed is a regular hexagon.

10. A regular polygon is both equilateral and equiangular.

11. Any regular polygon can be inscribed in a circle.

12. Many of the terms associated with circles are also used with regular polygons.

13. The center of a regular polygon is the center of the circumscribed circle.

14. The radius of the regular polygon is the distance from the center to a vertex. r

15. A central angle is an angle formed by two radii drawn to two consecutive vertices.

16. The measure of a central angle of a regular polygon with “n” sides is _____.

17. The apothem of a regular polygon is the distance from the center to a side.

18. r a Check out the apothem and radius.

19. r a Check out the apothem and radius.

20. r a Check out the apothem and radius.

21. r a Check out the apothem and radius.

22. r a Check out the apothem and radius.

23. The area of a regular polygon is equal to half the product of the apothem and the perimeter. Theorem 11-6 A regular polygon = ½ ap

24. Here is how that works … Take a regular n-gon and draw all its radii. That will produce “n” triangles. The area of one of those triangles is …

25. r s s s s s s r a

26. Take a regular n-gon and draw all its radii. That will produce “n” triangles. The area of one of those triangles is … Take this times the “n” triangles …

27. A regular triangle = ½ ap r a ss Also known as (a.k.a.): Equilateral Triangle Equiangular Triangle ½ s

28. Don’t miss this! r a ss 120º 60º 30º 60º 30º See the 30º-60º-90º ∆’s! 2a = ½ s 30º

29. A regular quadrilateral = ½ ap r a s s s a.k.a.: square ½ s

30. r a s s s Don’t miss this! ½ s 45º 90º 45º See the 45º-45º-90º ∆’s! a = = a 2a = = 2a 45º

31. A regular hexagon = ½ ap r a s s ss s ½ s

32. r a s s ss s Don’t miss this! ½ s 120º 60º 120º 60º 30º 60º

33. r a s s ss s Don’t miss this! ½ s 60º See the 30º-60º-90º ∆’s! 60º 30º 60º s = r = = r ½ r =

34. Example #1 Find the perimeter and area of a regular triangle with its apothem equal to 9 feet. 9’9’ 30º 60º

35. Example #2 Find the apothem and radius of a regular quadrilateral with an area of 100 sq. yd. 10 yd 5 yd 45º a = 5 yda r

36. Example #3 Find the area of a regular hexagon with a side equal to 12 units. 12u 6u 60º 30º

37. Assignment Written Exercises on pages 443 & 444 REQUIRED: 1 to 15 odd numbers * Bonus: 18, 19, 20 * How can you calculate the area of any figure? UPDATE YOUR STUDENT AID CARD!