Ratio of Area of Circle to Square

CAT 1995 Slot 1 · QA · Medium · Geometry

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PQRSPQRS is a square. SRSR is a tangent (at point SS) to the circle with centre OO and TR=OSTR = OS. Then the ratio of area of the circle to the area of the square is:

  1. A.

    \frac{\pi}{3}

  2. B.

    \frac{11}{7}

  3. C.

    \frac{3}{\pi}

  4. D.

    \frac{7}{11}

Answer

A

Explanation

Let radius of circle OS=rOS = r. TR=OS=r  ⟹  OR=OT+TR=r+r=2rTR = OS = r \implies OR = OT + TR = r + r = 2r. In right-angled triangle △OSR\triangle OSR, OS⊥SROS \perp SR. SR2=OR2−OS2=(2r)2−r2=3r2SR^2 = OR^2 - OS^2 = (2r)^2 - r^2 = 3r^2. Since PQRSPQRS is a square, side SR=3rSR = \sqrt{3}r, so Area of square =SR2=3r2= SR^2 = 3r^2. Area of circle =πr2= \pi r^2. Ratio =πr23r2=π3= \frac{\pi r^2}{3r^2} = \frac{\pi}{3}.

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