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?

Answer 1

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
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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