By: Tao Steven Zheng (郑涛)

【Problem】

Hippocrates of Chios (c. 470 – c. 410 BCE) was an ancient Greek mathematician who worked on the classical problems of “squaring the circle” and “doubling the cube”. The “Lune of Hippocrates” originates from Hippocrates’s attempt of squaring the circle. Let be the area of the lune (region shaded in orange), and  be the area of the right triangle (region shaded in blue). What is the ratio of the two areas?

【Solution】

Let  be the area of the lune, and  the area right triangle .  Since the right triangle is isosceles,   and  . Hence, the area of the right triangle is

To determine the area of the lune, subtract the area of the unshaded region between the lune and the right triangle from the semicircle of diameter .

Therefore, .