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Squares Circumscribed by Circles

When a square is circumscribed by a circle , the diagonal of the square is equal to the diameter of the circle.

Example 1:

Find the side length s of the square.

Math diagram

The diagonal of the square is 3 inches. We know from the Pythagorean Theorem that the diagonal of a square is 2 times the length of a side. Therefore:

s 2 = 3 s = 3 2 = 3 2 2 in .


Example 2:

Find the area of the circle.

Math diagram

First, find the diagonal of the square. Its length is 2 times the length of the side, or 5 2 cm.

This value is also the diameter of the circle. So, the radius of the circle is half that length, or 5 2 2 .

To find the area of the circle, use the formula A = π r 2 .

A = π ( 5 2 2 ) 2 = π ( 25 2 4 ) = 25 2 π cm 2


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