A circle and a square have the same area. What is the ratio of the area of a square inscribed in the circle to the area of a circle inscribed in the square? (Draw nice pictures.)
r2 = s2 → r2 = s2 / A1 = Asquare = half the product of the diagonals = ½(2r · 2r) = 2r2
A2 = Acircle =
(s/2)2 = s2 / 4A1 / A2 = (2r2) / (
s2 / 4)
= (2s2/) / (s2 / 4)
= 2s2/ · 4/s2
= 8/2
) / (s2 / 4)= 2s2/
· 4/s2= 8/
2