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

it would be 4x5x6x3x6

Answer 2

The ratio of the circumference of the circle to the perimeter of the square is
`\pi √(2):4`

Step-by-step explanation

Area of the square = 9 inch²

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

`a^2 = 9 \\ \\ a= √(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√(2) = 3√(2)` inch

So, the diameter of the circle
`= 3√(2)` inch

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

`2r = 3√(2) \\ \\ r= (3√(2))/(2)`

Circumference of the circle,
`2\pi r= 2\pi *(3√(2))/(2)= 3\pi √(2)` inch

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

So, the ratio will be:
`3\pi √(2) : 12 = \pi √(2) : 4`