## anonymous 5 years ago how do I figure out area of a polygon with perimeter of 108

1. anonymous

It depends. Is the polygon regular (that is, does it have equal internal angles and equal sides)?

2. xkehaulanix

Is it a regular polygon?

3. anonymous

It's a 9 sided polygon

4. anonymous

Okay, it sounds like it's regular, since 9 divides 108. If it's regular, you can look at the shape and break it up into 9 isosceles triangles. You can find the area using a combination of cosine rule (to find the other sides of the isosceles triangle) and then formula for area of a triangle (1/2 ab sinC). The angle C will be 360degrees/(number of sides). The formula for the area will be $A=\frac{na^2\sin (360^o/n)}{4(1-\cos(360^o/n))}$where n is the number of sides, a is the side length. Here, $a=\frac{108}{9}=12$ and n = 9, so$A=\frac{9 \times 12^2.\sin(40^o)}{4 \times (1-\cos(40^o))} \approx 890$square units. If you want a proper derivation of the formula, let me know. I have to rush right now. Also, the formula only holds for regular polygons; that is, polygons that have the same side length and equal internal angles.