Question

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

Answer 1

The ratio of the circumference of the circle to the perimeter of the square is $\pi \sqrt{2}:4$

Explanation

Area of the square = 9 inch²

If the side length of the square is $a$, then

$a^2 = 9 \\ \\ a= \sqrt{9} =3$

So the side length of the square is 3 inch.

Now as the square is inscribed in a circle, so the diagonal of the square will be diameter of the circle.

Length of the diagonal of square = $a\sqrt{2} = 3\sqrt{2}$ inch

So, the diameter of the circle $= 3\sqrt{2}$ inch

If the radius of the circle is $r$, then

$2r = 3\sqrt{2} \\ \\ r= \frac{3\sqrt{2}}{2}$

Circumference of the circle, $2\pi r= 2\pi *\frac{3\sqrt{2}}{2}= 3\pi \sqrt{2}$ inch

and Perimeter of the square, $4a = (4*3)inch= 12$ inch

So, the ratio will be: $3\pi \sqrt{2} : 12 = \pi \sqrt{2} : 4$

Answer 2

It would be 4x5x6x3x6