# Squares in Semicircle and Circle

Prove the following statement

A square inscribed in a semicircle has 2/5 the area of a square inscribed in a circle of the same radius.

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Copyright © 1996-2018 Alexander Bogomolny

### Proof

Taking the common radius of the circles to be √5, draw the circles centered in a node of a unit square grid as shown. On the right, the side of the square is a hypotenuse of a right triangle with legs 1 and 3 and therefore has a side length of √10 and the area of 10.

On the left the square has area of 4. The ratio of the two areas is therefore

### References

- C. Alsina, R. B. Nelsen,
*Charming Proofs*, MAA, 2010, p. 123

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Copyright © 1996-2018 Alexander Bogomolny