ABCD is a square. Γ1 is a circle that circumscribes ABCD (i.e. Γ1 passes through points A,B,C and D). Γ2 is a circle that is inscribed in ABCD (i.e. Γ2 is tangential to sides AB,BC,CD and DA). If the area of Γ1 is 100, what is the area of Γ2?
If the square has side s,
The diameter of Γ1 is the diagonal of length s√2.
So 100 = π/2 s^2
s = √(200/π)
The radius of Γ2 is s/2, so its area is
π * 50/π = 50
--------------------------
Doing it all algebraically, the ratio of the radii r/R of Γ2/Γ1 is
(s/2) / (s/√2) = 1/√2
So, the ratio of the areas is 1/2