Henry Ernest Dudeney/Modern Puzzles/116 - The Crescent and the Star/Solution

by : $116$

 * The Crescent and the Star

Solution
They are both the same area: $36$ square units.

Proof
There are $12$ instances of the small equilateral triangle that forms the point $D$.

Hence the area of the star is $12 \times 3 = 36$ square units.

From Lune of Hippocrates, the area of the crescent is equal to the area of the square whose diagonal equals $BC$.

The side of this square is half the distance $AB$, which is $6$.

Hence from Area of Square, the area of the crescent is equal to $6 \times 6$, that is, $36$ square units.