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If a square is inscribed in a circle, find the ratio of the areas of the circle and the square.

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Answer

Let a be the side of the square.

If a square is inscribed in a circle, find the ratio of the areas of the circle and the square. Area and Perimeter of Plane Figures, Concise Mathematics Solutions ICSE Class 9.

By using the Pythagoras theorem,

AB2 + BC2 = AC2

⇒ a2 + a2 = AC2

⇒ AC2 = 2a2

⇒ AC = a 2\sqrt{2}

Diagonal of the square = Diameter of the circle

d = a 2\sqrt{2}

Radius, r = d2\dfrac{d}{2} = a22\dfrac{a \sqrt{2}}{2}

Now, the ratio of the area of the circle to the area of the square is:

=Area of circleArea of square=πr2side2=π×(a22)2a2=π×a2×24a2=π×a22a2=π2=2214=117= \dfrac{\text{Area of circle}}{\text{Area of square}}\\[1em] = \dfrac{πr^2}{\text{side}^2}\\[1em] = \dfrac{π \times \Big(\dfrac{a \sqrt{2}}{2}\Big)^2}{a^2}\\[1em] = \dfrac{π \times a^2 \times 2}{4a^2}\\[1em] = \dfrac{π \times a^2}{2a^2}\\[1em] = \dfrac{π}{2}\\[1em] = \dfrac{22}{14}\\[1em] = \dfrac{11}{7}\\[1em]

Hence, the ratio of the area of the circle to the area of the square is 11 : 7.

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