Question

A square is inscribed in a circle. If the area of the square is 9 in2, what is the ratio of the radius of the circle to the side of the square?

Answer 1

The ratio of the radius of the circle to the side of the square is 1 : 2.

Step-by-step explanation:

The ratio of the radius of the circle to the side of the square can be determined by finding the dimensions of the square and then calculating the radius of the circle using the formula A = πr².

Given that the area of the square is 9 in², we can find the side length of the square by taking the square root of the area: side length = √9 = 3 in.

To find the radius of the circle, we divide the side length of the square by 2: radius = 3/2 = 1.5 in.

Therefore, the ratio of the radius of the circle to the side of the square is 1.5 in : 3 in, or 1 : 2.

Answer 2

Regardless of the size of the square, half the diagonal is (√2)/2 times the side of the square.

The ratio is (√2)/2.

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Consider a square of side length 1. The Pythagorean theorem tells you the diagonal measure (d) is ...

... d² = 1² +1² = 2

... d = √2

The distance from the center of the square to one of its corners (on the circumscribing circle) is then d/2 = (√2)/2. This is the radius of the circle in which our unit square is inscribed.

Since we're only interested in the ratio of the radius to the side length, using a side length of 1 gets us to that ratio directly.