math
posted by Anonymous on .
area of a nonagon with a perimeter of 126in

The area A of any regular polygon of n sides with side length s, apothem a is
A=nas/2.
Since ns=perimeter, p, we can also write the area as
A=ap/2
An apothem is the distance from the centre of the polygon to the side.
We assume the given nonagon is regular, i.e. all sides and interior angles are equal.
A nonagon has 9 sides. Each side is of length 126/9=14". We need first to find the apothem as follows.
The central angle for each side is 360/9=40°.
Each side of the nonagon forms an isosceles triangle with the centre of the nonagon. Its half vertical angle equals 40/2=20°.
The height of this triangle is the apothem and can be calculated by the trigonometric formula
a=(s/2)/tan(20)=7cot(20)
Using the formula above:
Area = ap/2=126*7cot(20)=882cot(20°)