Two sectors of a circle of radius 12 overlap as shown, with P and R as the centers of the respective circles. Determine the area of the shaded region.
Dividing the shaded area in half from the bottom left vertex of the figure to the point of intersection we have this area :
Area of a quarter circle with a radius of 12 - area of a right triangle with legs of 12 =
pi (12)^2 / 4 = (1/2) (12)^2 = (12)^2 ( pi/4 - 1/2)
And by symmetry the total shaded area =
2 (12^2) ( pi/4 - 1/2) = 288 ( pi/4 - 1/2) units^2 ≈ 82.19 units^2