Calculate the area enclosed by the inscribed and circumscribed circles to a square with a diagonal of $8$ m in length.
- $16.89\ m^2$
- $25.13\ m^2$
- $15.64\ m^2$
-
none of these
For a square with diagonal d = 8 m, the circumscribed circle has a diameter equal to the diagonal, so its radius R is 4 m. The inscribed circle has a diameter equal to the side of the square, which is 8 / sqrt(2) m, so its radius r is 2 * sqrt(2) m. The enclosed area is the difference between the two circle areas, which is pi * (R^2 - r^2) = pi * (16 - 8) = 8 * pi, approximately 25.13 square meters.
The side of the square is found using the Pythagorean theorem, s squared plus s squared equals 8 squared, meaning 2 times s squared equals 64 and the side length is the square root of 32. The radius of the circumscribed circle is half the diagonal, which is 4, giving an area of pi times 4 squared or 16 pi. The radius of the inscribed circle is half the side, which is the square root of 32 divided by 2, giving an area of pi times the square root of 8 squared, resulting in 8 pi. The difference between the areas is 16 pi minus 8 pi, which equals 8 pi or approximately 25.13 square meters.