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):
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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