area of a regular polygon with 100 sides and a perimeter of 100 units

The area is 795.5 sq units.

We first find the apothem, using the formula:
[tex]a=\frac{s}{2\tan{(\frac{180}{n})}}[/tex]

Since there are 100 sides, all sides are equal and the perimeter is 100, the side length, s, is 1:
[tex]a=\frac{1}{2\tan{(\frac{180}{100})}}=15.91[/tex]

Now we find the area of the polygon.  Since there are 100 sides, there will be 100 equal triangles using the center as a vertex.  The formula for the area of a triangle is A=1/2bh; in this case, b is s, the side length, and h is a, the apothem. This gives us:
A=100(1/2)(s)(a) = 100(1/2)(1)(15.91) = 795.5.