Ratio of Inscribed to Circumscribed Circle Areas

CAT 1991 Slot 1 · QA · Easy · Geometry

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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. A.

    2 : 3

  2. B.

    3 : 4

  3. C.

    1 : 4

  4. D.

    1 : 2

Answer

D

Explanation

Let the side of the square be aa.

  • Radius of inscribed circle r=a2r = \frac{a}{2}, Area A1=π(a2)2=πa24A_1 = \pi \left(\frac{a}{2}\right)^2 = \frac{\pi a^2}{4}.
  • Radius of circumscribed circle R=a22R = \frac{a\sqrt{2}}{2}, Area A2=π(a22)2=πa22A_2 = \pi \left(\frac{a\sqrt{2}}{2}\right)^2 = \frac{\pi a^2}{2}.
  • Ratio A1A2=12\frac{A_1}{A_2} = \frac{1}{2}, i.e., 1:21 : 2.

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