Chapter 14

1. Chapter 14 By Amalia Collins

2. Lesson 1

3. Definitions and Theorems • Regular Polygon: a convex polygon that is both equilateral and equiangular • Apothem: a perpendicular line segment in a regular polygon that goes from its center to one of its sides • Theorem 74: Every regular polygon is cyclic.

4. Important Information • A polygon is inscribed in a circle iff all of its vertices lie on that circle. • The circle is circumscribed about this polygon. • A polygon and its circumscribed circle have the same center. • An angle formed by radii drawn to two consecutive vertices of a regular polygon is called a central angle.

5. Some examples of regular polygons. *some additional examples of regular polygons include but are not limited to: regular heptagons, regular nonagons, regular decagons, and regular dodecagons

6. All regular polygons are cyclic.

7. Regular polygons have the same center as their circumscribed circle.

8. central angle radii apothem

9. Lesson 2

10. Theorem 75 • The perimeter of a regular polygon having n sides is 2Nr, in which N = n sinand r is its radius. • In other words… p = 2(n sin)r *The more sides a polygon has, the closer its perimeter is to that of the circle in which it is inscribed and the closer N is to π.

11. 8 cm r = 8 cm n = 5

12. p = 2Nr and N= n sin N= 5sin so p = 2(5 sin)•8 p ≈ 47.02 cm

13. Lesson 3

14. Theorem 76 • The area of a regular polygon having n sides is Mr2, in which M = n sincosand r is its radius. • In other words… A = (n sin cos)r2 *The more sides a polygon has, the closer its area is to that of the circle in which it is inscribed and the closer M is to π.

15. 8 cm r = 8 cm n = 5

16. A = Mr2 and M= n sincos M = 5sincos so A = 5 sin•82 A ≈ 188.1 cm2

17. Lesson 4

18. Definitions and Theorems • Circumference: the limit of the perimeters of a circle's inscribed regular polygons • Theorem 77: If the radius of a circle is r, its circumference is 2πr. • Corollary: If the diameter of a circle is d, its circumference is πd. • Note that π =

19. 3 ft C = 2πr C = 2π•3 C = 6π ft ≈ 18.85 ft

20. Lesson 5

21. Definitions and Theorems • Area of a Circle: the limit of the areas of the circle’s inscribed regular polygons • Theorem 78: If the radius of a circle is r, its area is πr2.

22. 3 ft A = πr2 A = π32 A = 9π ft2≈ 28.27 ft2

23. Lesson 6

24. Key Information • Sector: a region of a circle bounded by an arc of the circle and the two radii to the endpoints of the arc • The area of a sector is πr2, where m is the measure of its central angle. • The length of an arc is 2πr.

25. More Useful Information • If a sector is a certain fraction of a circle, then it’s area is the same fraction of the circle’s area. • Similarly, if an arc is a certain fraction of a circle, then it’s length is the same fraction of the circle’s circumference.

26. Example • If this sector is ¼ of the circle, and its arc is ¼ of the circle, and the circle has an area of 40 units2and a circumference of about 22.42 units… • The sector has an area of 10 units2, and an arc length of about 5.6 units. arc sector