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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.

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    $\begingroup$ Do you mean a regular polygon? There many quadrilaterals all with equal sides but not equal angles. $\endgroup$ – lhf Apr 21 '12 at 15:06
  • $\begingroup$ See Wikipedia $\endgroup$ – lhf Apr 21 '12 at 15:06
  • $\begingroup$ Perhaps you means perimeter of the polygon. $\endgroup$ – agtortorella Apr 21 '12 at 15:07
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    $\begingroup$ You can make a rhombus of arbitrarily small area with four equal sides. I assume you are interested in regular polygons. I would suggest drawing a diagram and joining all the vertices to the centre. If you know how to compute the area of a triangle given two sides and the angle between them, you should be able to complete the task. $\endgroup$ – Mark Bennet Apr 21 '12 at 15:07
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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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Use The equation: $A = .5asn$

This works with all regular polygons.

$a$ is the length of the apothem(Perpendicular bisector of one side to the center point of the polygon)

$s$ is the length of each side

$n$ is the number of sides

This is really is splitting the polygon into triangles where the apothem is the height of said triangle and each side of the polygon is the base of that triangle. $n$ is just the number of sides/number of triangles

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One way to get these values is by calculating the length $c$ of the sides of your regular n-gon. Asumme the radius of the circumcircle is $r$. Then each side of the $n$-gon is the side of a triangle with third point the middle of the circumcircle and length $r, r$ and $c$. The angle bounded by the to sides of length $r$ is $2\pi/n$, so by the law of cosines you have $$ c^2 = r^2+r^2 - 2r^2 \cos(\frac{2\pi}{n}) = 2r^2(1-\cos(\frac{2\pi}{n}))$$ From this you can calculate $c$ and with $nc$ you know the circumference. The area is just $n$ times the area of the triangle, for which there are many well known formulas.

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