A circle is circumscribed and inscribed in a square of area 64cm. Calculate the area of the circular ring bounded by these two circles.

1. Given that the area of the square is 64 cm\(^2\), so the side length \( s \) of the square is given by:

s^2 = 64 \implies s = 8 \, \text{cm}

2. The circumscribed circle has radius \( R \) equal to half the diagonal of the square, which is equal to the side length times the root of 2,

R = \frac{8\sqrt{2}}{2} = 4\sqrt{2} \, \text{cm} 2. The area of the circumscribed circle:

A_{\text{circumscribed}} = \pi (4\sqrt{2})^2 = 32\pi \, \text{cm}^2

2. The inscribed circle has radius \( r \) equal to half the side length of the square,

r=\frac{8}{2}=4\text{cm}

2. The area of the inscribed circle:

A_{\text{inscribed}}=\pi(4)^2=16\pi\,\text{cm}^2

3. The area of the ring:

A_{\text{ring}} = A_{\text{circumscribed}} - A_{\text{inscribed}} = (32\pi - 16\pi) = 16\pi \, \text{cm}^2

Answer: 16\pi\operatorname{cm}^2