Puzzle #5

March 19, 2025

Two squares meet at the center/centre of a circle. The yellow square has area 20.

What is the area of the red square?

This puzzle was published on The Guardian.

Solution

Let r be the radius of the circle.

As the area of the yellow square is 20, its side length is

$\sqrt{20}=2\sqrt{5}$

Using the Pythagorean/Pythagoras’ theorem in this square, we get:

$r^2=\left(\sqrt{5}\right)^2+\left(2\sqrt{5}\right)^2$

$r^2=5+20$

$r^2=25$

$r=5$

The red square has its diagonal length 5. Let x be its side length.

Using the Pythagorean/Pythagoras’ theorem in this square, we get:

$5^2=x^2+x^2$

$25=2x^2$

$x^2=12.5$

So, the area of the red square is 12.5.

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