SOLUTION: A square is inscribed in a circle. If the area of the square is 9 in^2, what is the ratio of the circumference of the circle to the area of the circle? me and my in-law can't so

Question 1064339: A square is inscribed in a circle. If the area of the square is 9 in^2, what is the ratio of the circumference of the circle to the area of the circle?
me and my in-law can't solve it. i tried many different solutions and none of them are correct. it would be great if u can help me
Found 2 solutions by josgarithmetic, Boreal:
Answer by josgarithmetic(39617) (Show Source): You can put this solution on YOUR website!
Diagonal of the square is equal to diameter of the circle.

Ratio of circumference of the CIRCLE to the area of the CIRCLE:

---------For ANY circle.

The square and circle that you and your in-law has, the square is 9 inches^2, so side length is 3.
The diagonal, .
Radius .

For this diagonal or diamete , the ratio of circumference to area is

Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
A square inscribed in a circle has right angles which subtend 180 degrees of arc. Therefore, the diagonal of the square is the diameter of the circle. If the square is 9 in ^2 in area, each side is 3 inches, and the diameter is 3 sqrt (2) inches, and the radius (3/2) sqrt (2) inches. We get the diameter by the diagonal of a square, which is a 45-45-90 right triangle.
The area of the circle is pi*(3/2)sqrt(2)^2=pi(9/4)*2=(9pi/2) square inches.
The circle's circumference is pi*D=3 sqrt (2)*pi
The ratio of 3 sqrt (2)* pi/(9/2) pi=3 sqrt(2)*2/9=2 sqrt(2)/3
Any circle has an area of pi*r^2 and circumference pi*D or 2*pi*r
The ratio of 2pi*r/pi*r^2=2/r.
Using (3/2)sqrt (2) for r, we have 2/(3/2)sqrt(2)=4/3sqrt(2).
Rationalize the denominator and we get 4 sqrt(2)/6=2 sqrt(2)/3
2 sqrt (2)/3 is the answer, and that ratio is 0.943 numerically.
Note that the units are in^(-1), because the circumference is inches and the area in^2.

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