how do you find the length of a side on a regular hexagon only using the degrees of its angle?

Question

for example I have a circle with a radius of 6 inside a regular hexagon with all the angles equaling 120 degrees. I need to find the side of the hexagon with only that information.

Answer 1

First draw all the diagonals, forming 6 equilateral triangles, which have angles of 60 degrees. Divide one of them in half, and use trigonometry to determine one of the sides, as follows:

Tan(30) = x / 6 = 0.577

That gives you x = 3.464, or 2x = 6.928, which is the lengh of the side.

Tan(30) can be derived through Pythagorean's Theorem, which gives
Tan(30) = 1/Sqrt(3) = 0.577

If the circle is oustide of the hexagon, then the sides equal 6.

Answer 2

it will be equal to the radius.