100-gon with apothem 4

Question

anonymous · Answer

$$\text{Area}=\frac{\text{(apothem)(perimeter)}}{2}$$
I'm going to use a drawing here, but there's no way I'll draw 100 sides. I'll keep it simple and use a hexagon. Basically, given any regular $n$-gon, you can deconstruct it into $n$ congruent triangles, where the base of each triangle is the side length of the $n$-gon and the height is the apothem.
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In order to use the formula above for area, we must first find the perimeter. We can use trig to figure out the side length and get the perimeter from there.

Recall that the exterior angles of a regular $n$-gon add up to 360 degrees. This means any individual exterior angle has measure $\dfrac{360}{100}=3.6$ degrees.

Every interior angle is supplementary to every exterior angle, i.e. $\text{ext}+\text{int}=180^\circ$. So, each interior angle has measure $180-3.6=176.4$ degrees.

When you consider one of the many triangles you could draw in the 100-gon, the interior angle is bisected and you have the following:
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