Question

Find the area of the regular polygon. Give the answer to the nearest tenth. decagon with a side of 4 cm

Answer 1

Let n= the number of sides of a regular polygon
s=the length of each side.
A= the area of the n-gon

Divide the n-gon into n congruent isosceles triangles. We'll do it here for a 4-gon, where each side is 6cm.



Now divide the iso triangles into congruent right triangles.



Figure out the angle at the middle of each right triangle.



x = 360/(n/2)
= 180/n

using trig:

tan(x) = (s/2)/h

or h = s / 2tan(180/n)

the area of one of the right triangles is thus

a= (1/2)(s/2)(s / (2tan(180/n)))

a= s^2 / (8tan(180/n))

and the area of the n-gon is 2*n*a, so

A= s^2 / (4*tan(180/n))


I would suggest working through it with the square (n=4, s=6cm) if you are still having difficulty.


Good Luck