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Maths-

General

Easy

Question

The area of the circle and the area of a regular polygon of $n$ sides and of perimeter equal to that of the circle are in the ratio of

$\tan (\frac{π}{n}) : \frac{π}{n}$
$\cos (\frac{π}{n}) : \frac{π}{n}$
$\sin \frac{π}{n} : \frac{π}{n}$
$\cot (\frac{π}{n}) : \frac{π}{n}$

The correct answer is: $tan invisible function application open parentheses fraction numerator pi over denominator n end fraction close parentheses colon fraction numerator pi over denominator n end fraction$

Let $r$ be the radius of the circle and $A subscript 1 end subscript$ be its area
Then $A subscript 1 end subscript equals pi r to the power of 2 end exponent.$ Since the perimeter of the circle is same as the perimeter of a regular polygon of $n$ sides, therefore $2 pi r equals n a comma blank$ where $a$ is the length of one side of the regular polygon
$rightwards double arrow a equals fraction numerator 2 pi r over denominator n end fraction$
Let $A subscript 2 end subscript$ be the area of the polygon
Then, $blank A subscript 2 end subscript equals fraction numerator 1 over denominator 4 end fraction blank pi a to the power of 2 end exponent cot invisible function application open parentheses fraction numerator pi over denominator n end fraction close parentheses equals fraction numerator pi to the power of 2 end exponent r to the power of 2 end exponent over denominator n end fraction cot invisible function application open parentheses fraction numerator pi over denominator n end fraction close parentheses blank rightwards double arrow A subscript 1 end subscript blank colon A subscript 2 end subscript equals pi r to the power of 2 end exponent blank colon fraction numerator pi to the power of 2 end exponent r to the power of 2 end exponent over denominator n end fraction cot invisible function application open parentheses fraction numerator pi over denominator n end fraction close parentheses equals tan invisible function application open parentheses fraction numerator pi over denominator n end fraction close parentheses blank colon fraction numerator pi over denominator n end fraction$

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