The center of a circle is at a vertex of an equilateral triangle as shown. If the area of the circle and the equilateral triangle are both 10, what is the combined area of the two gray regions?
The grey shaded area = Area of equilateral triangle + Area of circle - 2(area of white sector)
Since the vertex of the equilateral triangle = 60° the also equals the central angle of the white sector ...so this white sector comprises 1/6 of the circle's area....= 10/6 = 5/3 unts^2
So...the grey area = 10 + 10 - 2(5/3) = 20 - 10/3 = [ 60 - 10] /3 = 50/3 units^2