# Square and Arc Puzzle

A smaller shaded square is wedged between two quarter circle arcs. What is the ratio of the smaller square to the larger square?

*What is the ratio of the smaller square to the larger square?*

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## Solution

We'll define the side of the smaller square as

*x*. Through symmetry, there is an equal gap either side, which we'll call*a*. The side of the larger square is, therefore*(x+2a)*.The yellow line is the radius of the circle, and is also length

*(x+2a)*.Using Pythagoras, we can derive an equation between

*x*and*a*:Using the standard quadratic formula:

The only sensible answer is the positive one.

The side of the smaller square is 3/5ths of the side of the larger square.

The ratio of the areas is:

## Result

*The area of the smaller square is 36% the area of the larger square*.