November 2025 Problem of the Month

Take a square of side length 1 unit and draw four quarter circles within the square, each with radius 1 unit and center at a corner of the square.

This divides the square into nine partitions with essentially three distinct shapes.

What is the area of each of the shapes created?

Solution to the Problem:

Here is the solution:

The areas of the three shapes:


Set up three equations with the three variables a, b, and c.
The area of the square is 1 square unit.
The center of each quarter-circle arc is a corner of the square.
Each quarter circle has an area = π / 4 square units.

For the first two equations, refer to the original diagram in the problem.

Equation #1:
  2a + 3b + c = π / 4
Equation #2:
  2a + b = 1 - π / 4

For a third equation, draw the equilateral triangle shown in red in the figure below:





Now check these answers in the equation showing the nine areas adding up to 1:

4a + 4b + c = 1
4(.043388523) + 4(.127824792) + .315146744 = ?
.173554092 + .511309968 + .315146744 = 1.000010804