A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

1. 2 : 3
2. 3 : 4
3. 1 : 2
4. 1 : 4

Answer

Let a be the side of the square.

The side of the square will be the diameter of the inscribed circle.

Radius of inscribed circle = a/2

Area of inscribed circle = π(a/2)² = 1/4 a²π

The diagonal of the square will be the diameter of the circumscribed circle.

Radius of circumscribed circle = √2a/2

Area of circumscribed circle = π(√2a/2)² = 1/2 a²π

The ratio of the area is 1 : 2

The correct option is C.