FEATURED
Beautiful Geometry 1
Two Triangles and a Circle
Two half squares, and a circle of radius 1. What’s the total shaded area?
SOLUTION
Consider the inscribed trapezoid ABCD. Let M be the intersection of its diagonals and let O be the center of the circle.
Since ABCD is isosceles, we have ∠CAB = ∠ABD = 45, and then ∠AMD = 90. Also, ∠AOD = 2∠ABD = 90. Now, applying the Pythagorean Theorem to △AMD and △AOD, we get:
S = 2S(ABM) + 2S(MCD) = AM^2 + MD^2 = AD^2 = AO^2 + OD^2 = 2.
