Question

A square is inscribed in a circle. How much larger is the area of the circle than the area of the square?

ratio form

Answer 1

π/2 : 1 or π : 2

Explanation:

In the attachments I have a picture of a square inscribed in a circle.

Call the side of the square a. Therefore, the area of the square is a^2.

We can also see that the diagonal of the square is the circle's diameter.

Using the Pythagorean Theorem we find that the diagonal/diameter of the circle is a√2. Thus the radius is that divided by 2, which is (a√2)/2.

We also know that the formula for a circle's area is πr^2, so the area of the circle is π((a√2)/2)^2 --> π(a^2)(2)/4 --> π(a^2)/2.

Hence the ratio of the circle to the square is π(a^2)/2 to a^2 which is π/2 to 1 or π to 2.