Above, you see the axe of Al-Hazan. It consists of two blades and the handle. We don't care about the handle. Each blade is bounded by two small quarter circle arcs and one larger one. The center of the larger quarter arcs is at point O. The distance from O to the edge of the blades is 1 unit. Your task is to find out how big the blades are. Specifically, what is the area of the red shaded regions?
Take a crack at solving the puzzle before scrolling down for my solution. There are several ways to approach the problem. I suspect mine is the laziest.
Al-Hazan in the title is an unfortunate misspelling of the name of an Arab mathematician and physicist, Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham al-Basri. al-Basri had some famous lunes shown below, but, to the best of my knowledge, no axe.
My solution:
Add 3 auxiliary lines as shown. Because the edge of the blade is a quarter circle, the central angle at point O is a right angle and the triangle is a right triangle.
The areas labelled A, B and C are circle segments bounded by a quarter circle and the associated chord. They are therefore similar. That is, they have the same shape.
A, B and C are similar shapes constructed on the sides and hypotenuse of a right triangle, so the area of A + area of B = area of C. The blade consists of C plus the red wedge shape inside the triangle. The triangle consists of the red wedge, plus A, plus B. So the area of the blade is equal to the area of the triangle. The triangle has sides of length 1, the distance from the center of the circle to the edge of the blade. Therefore the triangle's area is (1 x 1)/2 = 1/2. There are 2 blades, so their combined area is 1.