A circle is inscribed in a square and circumscribed about it. Find the radii of these circles, if their difference is 4 cm.

Let the radius of the larger circle be R cm, and the small circle r cm.

The radius of the small circle is equal to half the length of the square, then AB = 2 * r cm. From point O we draw a perpendicular OK to the side of the square AB. Then OK = r. Triangle BOK is rectangular and isosceles, OK = B. To = r. Then R ^ 2 = 2 * r ^ 2 or R = √2 * r. (1).

By assumption, R – r = 4 see (2).

Let’s solve the system of two equations 1 and 2.

R = 4 + r.

R = √2 * r.

4 + r = √2 * r.

(4 + r) ^ 2 = 2 * r ^ 2.

16 + 8 * r + r ^ 2 = 2 * r ^ 2.

r ^ 2 – 8 * r – 16 = 0.

Let’s solve the quadratic equation.

r = 4 + 4 * √2 cm.

Then R = 8 + 4 * √2 cm.

Answer: The radius of the larger circle is 8 + 4 * √2 cm, the smaller one is 4 + 4 * √2 cm.