Regular Polygon Area

Geometry Level 2

A regular polygon with 400 400 sides of length tan ⁡ 9 20 ∘ \sqrt{\tan{\frac{9}{20}}^{\circ}} has an area of x 2 , x^2, where x x is a positive integer. Find x x .


The answer is 10.

Alex Delhumeau by Alex Delhumeau , 20, USA

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1 solution

Alex Delhumeau
Oct 5, 2015

Start by converting 9 20 ∘ \frac{9}{20}^{\circ} into radians. It is π 400 \frac{\pi}{400} rad. Then apply the formula for the area of a regular polygon: A = 1 4 n s 2 cot ⁡ π n A = \frac{1}{4}ns^2\cot{\frac{\pi}{n}} where n n is the number of sides and s s is the side length.

⇒ 1 4 ∗ 400 ∗ tan ⁡ π 400 2 ∗ cot ⁡ π 400 = 100 . \Rightarrow \frac{1}{4}*400*\sqrt{\tan{\frac{\pi}{400}}}^2*\cot{\frac{\pi}{400}}=\boxed{100}.

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