# math

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area of a nonagon with a perimeter of 126in

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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°)

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