A square garden is to be built inside a circular area. Each corner of the square touches the circle. If the radius of the circle is 2, how much greater is the area of the circle than the square?

A square garden is to be built inside a circular area. Each corner of the square touches the circle. If the radius of the circle is 2, how much greater is the area of the circle than the square?

Detailed Explanation

Find the difference between the area of the circle and the area of the square. The area of the circle is \(πr^2 = π × 2^2 = 4π\). The area of the square is \(s^2\), where s represents the length of the square. The radius is half the length of the square’s diagonal, so the diagonal is 4. By the Pythagorean theorem, \(s^2 + s^2 = 4^2\) <=> \(2s^2 = 16\) so \(s^2 = 8\). The difference in area is 4π – 8.