What is the area of a regular decagon with radius 14 feet?

Question

What is the area of a regular decagon with radius 14 feet?  using trig.

Answer 1

A decagon has 10 sides and can be split into 10 triangle. We just need to find the area of one of those triangles and multiply by 10.

The vertex angle in the triangle will be 360/10 = 36 degrees. If we draw an altitude from the vertex to the base we will split the triangle into two equal right triangles with the top angle being 18 degrees and the hypotenuse being 14. Thus 20*the area of one of these right triangles will be the area of the whole decagon.

We height of the right triangle as follows:

cos(18) = height/14

height = 14*cos(18) = 13.31

We find the base of the right triangle as follows:

sin(18) = base/14

base = 14*sin(18) = 4.33

area pf right triangle = 1/2 * base * height = 1/2 * 13.31 * 4.33 = 28.82

area of whole decagon is 28.82 * 20 = 576.4

Answer 2

It has 10 sides so the angle will be 360 / 10 = 36°.

We need the area of 10 isosceles triangles, with angle 36° and two equal sides of 14 ft

Area of one will be 1/2 ab sin C with a=14, b=14 and C = 36°

So total area will be 10 x 0.5 x 14 x 14 x sin 36° which you can calculate