Geometry 10.5 Areas of Circles and Polygons

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

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

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    So, the area if 64 , or about square inches.

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  r  r  The diameter of the circle is about cm.

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)

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

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

1.Perimeter = = 42 m 2.Apothem: “ ” triangle = (7√3)/3 = Area = ½ Pa = ½ * 14 * 3 * 4 = 84 m²

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

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

A = ½ Pa If a = 5√3, than we have triangle so the side = 10 A = ½*6*10*5 √3 = 259.8cm²

A = ½ Pa If a = 7.5, and P = A = ½*7.5 * = m²

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

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