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If the circumference of a circle and the perimeter of a square are equal, then whose area is greater − that of circle or of square? |
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Answer» Let the radius of the circle is r and side length of a square is a. ∴ the circumference and the area of the circle are 2πr and πr2 respectively. And the perimeter and the area of the square are 4a and a2 respectively. Given that the circumference of the circle and the perimeter of the square are equal. ∴ 2πr = 4a ⇒ = \(\frac{2a}π\). Now, the ratio of the areas of the circle and area of the square is \(\frac{π r^2}{a^2}\) = \(\frac{(\frac{2a}π)^2}{a^2}\) = \(\frac{\frac{4a^2π}{π^2}}{a^2}\) = \(\frac{4 a^2}{a^2π}\) = \(\frac{4}π\) > 1. (∵ π = 3.14 < 4. ∴ 4/π > 1) Hence, \(\frac{Area\,of\,circle}{Area\,of\,square}\) > 1. ∴ Area of circle > Area of square. Hence, area of circle is greater. |
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