Two circles of radius 10 cm overlap such that each circle passes through the center of the other, as shown. What is the area of their overlap? Express your answer in the simplest radical form.
+128475 (1/2) of the gray area =
Area of a sector of 1/3 of the circle - area of an isosceles triangles with equal sides of 10 and an included angle of 120° =
(1/3) pi * 10^2 - (1/2) 10^2 sqrt (3) / 2 =
100 [ pi/3 - sqrt (3) / 4 ]
So....the whole gray area =
200 [ pi/3 - sqrt (3) / 4 ] = 200 [ 4pi - 3sqrt (3) ] / 12 = (50/3) ( 4pi - 3sqrt (3)) units^2 =
≈ 122.84 units^2
