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area of a nonagon with a perimeter of 126in

  • math - ,

    The area A of any regular polygon of n sides with side length s, apothem a is
    Since ns=perimeter, p, we can also write the area as

    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

    Using the formula above:
    Area = ap/2=126*7cot(20)=882cot(20°)

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