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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

  1. 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  
  2. cos invisible function application open parentheses fraction numerator pi over denominator n end fraction close parentheses colon fraction numerator pi over denominator n end fraction  
  3. sin invisible function application fraction numerator pi over denominator n end fraction colon fraction numerator pi over denominator n end fraction  
  4. cot invisible function application open parentheses fraction numerator pi over denominator n end fraction close parentheses colon fraction numerator pi over denominator n end fraction  

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 blankwhere 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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