2. Geometry 10.5 Areas of Circles and Polygons

3. Objectives Find the area of a circle and polygons To be able to solve problems with circles and polygons

4. Area of a Circle The area of a circle is  times the square of the radius or A =  r 2.

5. Using the Area of a Circle Find the area of P. Solution: a. Use r = 8 in the area formula. A =  r 2 =  8 2 = 64   201.06  So, the area if 64 , or about 201.06 square inches.

6. Using the Area of a Circle Find the diameter of Z. Solution: Area of circle Z is 96 cm 2. A =  r 2 96=  r 2 96/  = r 2 30.56  r 2 5.53  r  The diameter of the circle is about 11.06 cm.

7. The Apothem The apothem (a) is the segment drawn from the center of the polygon to the midpoint of the side (and perpendicular to the side)

8. More... a G F E DC B A H Hexagon ABCDEF with center G radius GA apothem GH

9. Area of a regular polygon The area of a regular polygon is: A = ½ Pa Area Perimeter Perimeterapothem

11. 45-45-90

15. 1.Perimeter = 14+14+14 = 42 m 2.Apothem: “30-60-90” triangle = (7√3)/3 = 4.0 3.Area = ½ Pa = ½ * 14 * 3 * 4 = 84 m²

16. 1.Perimeter = 15*4= 60 in 2.Apothem: “45-45-90” triangle = 7.5 in 3.Area = ½ Pa = ½ * 4*15*7.5 = 225 in²

17. Perimeter = 10*5= 50 m Apothem: “36-54-90” triangle = 5/tan 36 = 6.9 Area = ½ Pa = ½ * 50* 6.9 = 172.5 m² 1.Find central angle measure : 360/n = 360/ 5 = 72° 2.Apothem splits angle into halves 72/2 = 36 °

18. A = ½ Pa If a = 5√3, than we have 30-60-90 triangle so the side = 10 A = ½*6*10*5 √3 = 259.8cm²

19. A = ½ Pa If a = 7.5, and P = 54.49 A = ½*7.5 * 54.49 = 204.3 m²

20. A region = A square – A circle A square = 40*40 = 1600 A circle = 3.14*20² = 1256 A region = 1600-1256 = 344 m²

21. A region = A circle – A (2 smaller circles) A circle= 64*3.14 = 200.96 A one smaller circle = 16*3.14 = 50.24 A region = 200.96-2*50.24=100.5m²