A circle is described around the square, the length of which is 12 pi, and another circle is inscribed in the square, find its length

Knowing the length of the circumscribed circle, we determine its radius OA.

L1 = 2 * n * OA.

OA = L1 / 2 * n = 12 * n / 2 * n = 6 cm.

Then the diameter AB, which is equal to the diagonal of the square, will be equal to: AB = 2 * OA = 2 * 6 = 12 cm.

Since all sides of the square are equal, then from the right-angled triangle ABC, according to the Pythagorean theorem, AB ^ 2 = 2 * AC ^ 2.

AC ^ 2 = AB ^ 2/2 = 144/2 = 72.

AC = 6 * √2.

The side of the square is equal to the diameter of the inscribed circle, then the radius of the OS = AC / 2 = 6 * √2 / 2 = 3 * √2 cm.

Determine the length of the inscribed circle.

L2 = 2 * n * OS = 2 * n * 3 * √2 = 6 * n * √2 cm.

Answer: The length of the inscribed circle is 6 * n * √2 cm.