Find the area of a circle circumscribed about a regular quadrilateral with a perimeter of 24 cm.

A regular quadrilateral is a square, all sides of the square are equal. Knowing that the perimeter of the square is 24 cm, we can find its side:

a = P / 4 = 24/4 = 6 cm.

Two adjacent sides of a square and its diagonal form a right-angled triangle. By the Pythagorean theorem:

d ^ 2 = a ^ 2 + a ^ 2 = 2 * a ^ 2;

d = a√2 = 6√2 cm is the diagonal of the square.

The diameter of a circle circumscribed about a square is equal to the diagonal of this square, respectively, the radius of such a circle is half the diagonal:

r = d / 2 = 6√2 / 2 = 3√2 cm.

The area of the circle is determined by the formula:

S = πr2 = n * (3√2) 2 = n * 9 * 2 = 18n ≈ 56.52 cm2.