A circle is circumscribed around a square, and another circle is inscribed in the square. If the area of the square is 9 inches, what is the ratio of the circumference of the circumscribed circle to the circumference of the inscribed circle?

Answer 1

Let's start by finding the side length of the square. Since the area of the square is 9 inches, we know that the side length is √9 = 3 inches.

The diameter of the circumscribed circle is equal to the diagonal of the square, which can be found using the Pythagorean theorem. The diagonal of a square with side length a is a√2, so in this case, the diagonal is 3√2 inches. Therefore, the diameter of the circumscribed circle is 3√2 inches.

The radius of the circumscribed circle is half of the diameter, so it is 1.5√2 inches. The circumference of the circumscribed circle is 2π times the radius, so it is 3π√2 inches.

The radius of the inscribed circle is half of the side length of the square, so it is 1.5 inches. The circumference of the inscribed circle is 2π times the radius, so it is 3π inches.

Therefore, the ratio of the circumference of the circumscribed circle to the circumference of the inscribed circle is:

(3π√2) / (3π) = √2

So the ratio is √2:1.