Area of Polygons and Circles

1. Area of Polygons and Circles Chapter 11

2. 11.1 Angle Measures in Polygons • The sum of the measures of the interior angles of a polygon depends on the number of sides. • Determining how many triangles are in each polygon will help you figure out the sum of the measures of the interior angles.

3. Sum of interior angles • Draw all the diagonals from one vertex. This will divide the polygon into triangles.

4. Polygon Interior Angle Theorem • The sum of the measures of the interior angles of a convex n-gon is (n-2)*180 . • The measure of each interior angle of a regular n-gon is

5. Examples • Find the value of x.

6. Polygon Exterior Angle Theorem • The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360 . • The measure of each exterior angle of a regular n-gon is

7. Example • Find the value of x.

8. Example • Find the value of x.

9. 11.2 Areas of Regular Polygons • Regular Polygon: all sides are same length • You know that the area of a triangle is equal to A = ½ bh. • If you are dealing with an equilateral triangle there is a special formula: • A = • (s = side)

10. Example • Find the area of an equilateral triangle with 8-inch sides.

11. Vocabulary • When dealing with a polygon, think of it as if it were inscribed in a circle:

12. Vocabulary • Center of a polygon: the same as the center of the circumscribed circle • Radius of the polygon: the same as the radius of the circumscribed circle • G is the center of the polygon • GA is the radius F A G E B D C

13. Vocabulary • Apothem of the polygon: the distance from the center to any side of the polygon. • The apothem is the segment GH. F A H G E B D C

14. Area of a Regular Polygon • The area of a regular n-gon with side length s is half the product of the apothem a and the perimeter P. • A = 6 4

15. Central angle of a regular polygon • An angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. • You can divide 360 by the number of sides (n) to find the measure of each central angle.

16. Examples • Find the area of the regular octagon. • P = __________ • Apothem = ___________ • Area = _____ 8.3 4.3

17. 11.3 Similar Figures • If two polygons are similar with the lengths of corresponding sides in the ratio of a:b, then the ratio of their areas is a2:b2

18. Similar Figures • The ratio of the lengths of corresponding sides is 1:2. • The ratio of the perimeters is also 1:2. • The ratio of the areas is 1:4.

19. 11.4 Circumference and Arc Length • Circumference of a circle: the distance around the circle. • Arc length: a portion of the circumference of a circle. • Measure of an arc – degrees • Length of an arc – linear units • The circumference C of a circle is: • . • . • d is the diameter of the circle • r is the radius of the circle

20. Arc Length Corollary • The ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360 . • Arc length of AB = • Arc length of AB = A P B

21. Arc Lengths • The length of a semicircle = ½ of the circumference. • The length of a 90 arc = ¼ of the circumference.

22. Examples • Find the length of the arc.

23. Examples • Find the length of the arc.

24. Example • Find the circumference.

25. Examples • Find the measure of XY.

26. 11.5 Areas of Circles and Sectors • Area of a Circle = • Find the area of the circle. 8 in.

27. Examples • Find the diameter of the circle if the area is 96 cm2. Z

28. Sector of a Circle • Sector of a Circle: the region bounded by two radii of the circle and their intercepted arc.

29. Area of a Sector • The ratio of the area, A, of a sector of a circle to the area of the circle is equal to the ratio of the measure of the intercepted arc to 360 . • .

30. Examples • Find the area of the sector.

31. Examples • A and B are two points on the circle with radius 9 inches and m APB = 60 . Find the areas of each sector.

32. Finding areas of Regions = - Area of shaded region Area of circle Area of polygon

33. Examples • Find the area of the shaded region.

34. 11.6 Geometric Probability • Probability is a number from 0 to 1 that represents the chance that an event will occur. • Geometric Probability is a probability that involves a geometric measure such as length or area.

35. Probability and Length • Let AB be a segment that contains the segment CD. If a point K on AB is chosen at random, then the probability that it is on CD is: • P(Point K is on CD) = Length of CD Length of AB A C D B

36. Probability and Area • Let J be a region that contains region M. If a point K in J is chosen at random, then the probability that it is in region M is: • P(Point K is in region M) = Area of M Area of J J M

37. Examples • Find the probability that a point chosen at random on RS is on TU. • Find the probability that a point chosen at random on RS is on TU. R T U S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

38. Examples • Find the probability that a randomly chosen point in the figure lies in the shaded region.

39. Examples • Find the probability that a randomly chosen point in the figure lies in the shaded region.