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Nitya Nigam A lot of competition maths problems involve finding ratios of areas of various shapes, so having a quick way to find them is a useful tool. In this article, we'll derive the following equation for the area of a regular polygon: where s is the side length and n is the number of sides. To get to this equation, we first need to consider one of the n triangles that make up the polygon, one of which is shown in red on the right. The central angle of this triangle will be 360/n for any n-sided regular polygon. The green angle is 360/n, as stated above, so the orange angle θ is half of that, so θ = 180/n. tan θ = (s/2)/h so h = s / (2 tan θ). The area of a triangle is 1/2 of its base times its height, so the area of this triangle is 1/2 * s * s / (2 tan θ) = s^2 / (4 tan θ). We can then substitute θ = 180/n in, giving A_(triangle) = s^2 / (4 tan (180/n)).
Since there are n of these triangles in an n-sided regular polygon, the total area of the polygon is n * A_(triangle) = n s^2 / (4 tan (180/n)), as above.
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