SOLUTION: A circle is inscribed in a given square and another circle is circumscribed about the same square. If the area of the circumscribed circle is 4, what is the area of the inscribed c

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Question 255269: A circle is inscribed in a given square and another circle is circumscribed about the same square. If the area of the circumscribed circle is 4, what is the area of the inscribed circle?

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A circle is inscribed in a given square and another circle is circumscribed about the same square.
If the area of the circumscribed circle is 4, what is the area of the inscribed circle?
:
Find the radius of the outer circle
pi%2Ar%5E2+=+4
r%5E2+=+4%2Fpi
r = sqrt%284%2Fpi%29 is the radius of the large circle
then
diameter = 2sqrt%284%2Fpi%29
:
This is also the diagonal of the square
:
Find the square dimensions using this as the diagonal(a^2 + b^2 = c^2)
Find the sides of the square. (s = the side of the square)
s^2 + s^2 = (2sqrt%284%2Fpi%29)^2
2s^2 = 4(4/pi)
s^2 = 2(4/pi)
s^2 = 8/pi,
s = sqrt%288%2Fpi%29
s = 2sqrt%282%2Fpi%29 is the side of the square
:
This is also the diameter of the small circles, therefore
r = sqrt%282%2Fpi%29 (half the diameter)
r^2 = %282%2Fpi%29
Find the area of the small circle
A = pi+%2A+%282%2Fpi%29
A = 2 sq/cm; area of the small circle