SOLUTION: The length of a piece of wire is 8 inches. It is then formed into a regular hexadecagon (polygon of 16 sides). What is the area of the polygon?

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Question 1077071: The length of a piece of wire is 8 inches. It is then formed into a regular hexadecagon (polygon of 16 sides). What is the area of the polygon?
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39799) About Me  (Show Source):
You can put this solution on YOUR website!
The center and any two neighboring vertices of the polygon form an isosceles triangle, base side of length 1%2F2 inch. Apex angle measure 360%2F16=22%261%2F2 degrees; each base angle 78%263%2F4 degree each.

Length h, of either of the equal sides of one of these triangles:
sin%2822.5%29%2F%281%2F2%29=sin%2878.75%29%2Fh

h%2Fsin%2878.75%29=%281%2F2%29%2Fsin%2822.5%29

h=sin%2878.75%29%2F%282sin%2822.5%29%29


The ALTITUDE of one of these isosceles triangles:
a, the altitude
a%5E2%2B%28%281%2F2%29%2F2%29%5E2=h%5E2

a%5E2%2B%281%2F4%29%5E2=%28sin%2878.75%29%2F%282sin%2822.5%29%29%29%5E2

a%5E2=%28sin%2878.75%29%2F%282sin%2822.5%29%29%29%5E2-1%2F16

a=sqrt%28%281%2F4%29%28sin%2878.75%29%2Fsin%2822.5%29%29%5E2-1%2F16%29--------the altitude.
NOT finished.
Simplify this and then calculate the area for the whole polygon of 16 sides.


Area, 16%281%2F2%29%281%2F2%29%2Aa-----substitute for a, and simplify this expression.

Answer by MathTherapy(10809) About Me  (Show Source):
You can put this solution on YOUR website!

The length of a piece of wire is 8 inches. It is then formed into a regular hexadecagon (polygon of 16 sides). What is the area of the polygon?
Apothem = matrix%281%2C2%2C+.25%2Ftan+%2811.25%29%2C+inches%29

Area of one of the hexadecagon's triangles = matrix%281%2C3%2C+.5%28.5%29%28.25%2Ftan+%2811.25%29%29%2C+sq%2C+inches%29

Area of hexadecagon: