Almost the complete circle

Geometry Level 4

The Ashoka chakra situated in the middle of the Indian flag, has 24 spokes. If the coordinates of their tips were plotted in the argand plane such that they are the roots of the equation z 24 = 1 z^{24} = 1 , then find the area of the 24-sided polygon having the tips of the spokes as their vertices in consecutive order.

  • The answer is of the form a b ( a − 1 ) a\sqrt{b}\left(\sqrt{a} - 1\right) , submit the value of a + b a + b .
  • Consider the principal roots of the equation given in the question.


The answer is 5.

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Ashish Menon by Ashish Menon , 20, India

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1 solution

Ashish Menon
Dec 22, 2016

The circum radius of the resulting polygon would be 1 1 .
Then the area of the regular polygon thus formed is given by 1 2 n R 2 sin ⁡ ( 2 π n ) = 1 2 × 24 × 1 2 × sin ⁡ ( 2 π 24 ) = 3 2 ( 3 − 1 ) \dfrac{1}{2} nR^2 \sin\left(\frac{2\pi}{n}\right)\\ = \dfrac{1}{2} × 24 × 1^2 × \sin\left(\frac{2\pi}{24}\right)\\ = 3\sqrt{2}\left(\sqrt{3} - 1\right) .

⟹ a + b = 5 \implies a + b = \color{#3D99F6}{\boxed{5}} .

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You may change blue to Saffron and Green to the Left and Right of the Chakra

Vijay Simha - 2 years, 1 month ago

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