SOLUTION: A circle is inscribed in a square, which is circumscribed by another circle. If the diagonal of square is 2x, find the ratio of the area of the large circle to the area of the sma

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Question 347856: A circle is inscribed in a square, which is circumscribed by another circle. If the diagonal of square is 2x, find the ratio of the area of the large circle to the area of the small circle?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A circle is inscribed in a square, which is circumscribed by another circle.
If the diagonal of square is 2x, find the ratio of the area of the large
circle to the area of the small circle?
;
x = the radius of the large circle
then
A = pi%2Ax%5E2; the area of the larger circle
:
Find the area of the small circle
:
Let s = side of the square, given that 2x = diagonal of the square
s^2 + s^2 = (2x)^2
2s^2 = 4x^2
Divide both sides by 2
s^2 = 2x^2
s = sqrt%282x%5E2%29
s = x%2Asqrt%282%29 is the side of the square
:
the side of the square is also the diameter of the small circle,therefore:
%28x%2Asqrt%282%29%29%2F2 = the radius of the small circle
Find the area of the small circle
A = pi%2A%28x%2Asqrt%282%29%2F2%29%5E2
Which is:
A = pi%2A%28%282x%5E2%29%2F4%29
Cancel 2
A = pi%2A%28x%5E2%2F2%29
:
large area
---------- would be:
small area
%28pi%2Ax%5E2%29%2F%28pi%2A%28x%5E2%2F2%29%29 = 1%2F%281%2F2%29 = 2%2F1; canceled pi*x^2