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?
;
x = the radius of the large circle
then
A = {{{pi*x^2}}}; 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(2x^2)}}}
s = {{{x*sqrt(2)}}} is the side of the square
:
the side of the square is also the diameter of the small circle,therefore:
{{{(x*sqrt(2))/2}}} = the radius of the small circle
Find the area of the small circle
A = {{{pi*(x*sqrt(2)/2)^2}}}
Which is:
A = {{{pi*((2x^2)/4)}}}
Cancel 2
A = {{{pi*(x^2/2)}}}
:
large area
---------- would be:
small area

 {{{(pi*x^2)/(pi*(x^2/2))}}} = {{{1/(1/2)}}} = {{{2/1}}}; canceled pi*x^2