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? 
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Find the radius of the outer circle
{{{pi*r^2 = 4}}}
{{{r^2 = 4/pi}}}
r = {{{sqrt(4/pi)}}} is the radius of the large circle
then
diameter = {{{2sqrt(4/pi)}}} 
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This is also the diagonal of the square
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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(4/pi)}}})^2
2s^2 = 4(4/pi)
s^2 = 2(4/pi)
s^2 = 8/pi, 
s = {{{sqrt(8/pi)}}}
s = {{{2sqrt(2/pi)}}} is the side of the square
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This is also the diameter of the small circles, therefore
r = {{{sqrt(2/pi)}}} (half the diameter)
r^2 = {{{(2/pi)}}}
Find the area of the small circle
A = {{{pi * (2/pi)}}}
A = 2 sq/cm; area of the small circle