[Math] the area of a polygon with n equal length sides

geometry

What is the area and circumference of a polygon with n equal sides? (triangle, square, pentagon all the way to a circle)

It doesn't matter if it's based on the radius (let's call it r) or the length n.

EDIT: I ment regular polygon.

Best Answer

Assuming the polygon is regular ...

HINT: Cut it into $n$ isosceles triangles. Then compute the area of each of those triangles, and add them up.

SOLUTION:

Now you know the area formula for a triangle, in particular an isosceles triangle. If the top angle is $\theta = \frac{2\pi}{n}$ and the side length is $r$, then the height is $r\cos{(\theta/2)}$ and the base is $2r\sin{(\theta/2)}$, so the area of the isosceles triangle is $r^2\cos{(\theta/2)}\sin{(\theta/2)}$.

This makes the area of the polygon $nr^2\cos{(\theta/2)}\sin{(\theta/2)}$.

For the perimeter, add the bases up: $2nr\sin{(\theta/2)}$.

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