Squaring the Circle (Middle School)

"Squaring the Circle" is a problem proposed by the ancient Greeks.  The goal of the exercise was to draw a circle and a square of the same areas by using just a ruler and a compass.  It was shown in the 1800's to be unsolvable, but the exercise below made a lot of people think that a proof was possible.

Crescent of Hippocrates

Exercise: Find the area of the uppermost crescent in the Figure below, and show that it is not dependent on π.

Answer: if each side of the square is R, then the total area of the orange half-circle at the top must equal  π*R² /2
The area of the smaller brown sub-crescent must be subtracted from the above amount.
The sub-crescent's area equals the area of the quarter-circle formed by the sub-crescent and the triangle below it, less the area of the triangle.
The quarter-circle has radius R*√2 , and so has area

 π * (R*√2 )²/4 =π * 2 * R²/4 = π * R² /2

The area of the white and Orange triangle is the same as the area of the orange box at bottom right, or R².

The brown sub-crescent area must then be: (π * R² /2) - R².

The area of the uppermost crescent must then equal:

 (π * R² /2) - [(π * R² /2) - R²] = R², exactly the same as the area of the orange square.

Does it surprise you that no "pi" is involved in the area of this crescent? It surprised a lot of people when it was discovered, and led to a long (and fruitless) chase for ways to "square the circle".

Acknowledgements:

This problem was submitted by Howie Johnson of Twisp, Washington.