Determine the Radius of a Circle

[stanek]

Assume we are creating a circle around a regular polygon of N-sides, with the corners of the polygon lying on the circle.

ie.

2 sides is the diameter of the circle
3 sides is an equilateral triangle within the circle,
4 sides is a square within the circle, etc...
Just the basic shapes, nothing fancy.

Now assume that each line segment of the polygon is of a fixed length we will call X. So as we add more sides to the polygon the size of the circle will grow. My question is How do we determine the radius of the circle for N sides?

I can figure out 2 on my own which is 1/2X. yay for me
I am pretty sure 4 is equal to sqrtX.
I think 8 is 2sqrtX.

But im stumped on 3 through infinity. I feel like the answer should be simple and fairly obvious but I can't figure it out so I thought I'd ask. Thannks

[Borodog]

Law of sines.

[mickeyg13]

To elaborate on his answer, the central angle of the regular n-gon will be 2pi/n radians. Then consider an isosceles triangle where the radii form two legs with an angle of 2pi/n radians in between them. Then according to the Law of Sines, X/sin(2pi/n) = r/(sin theta), where theta is the one of the other angles of your triangle, namely 1/2*(pi - 2pi/n). Then r should be equal to X*sin(theta)/sin(2pi/n). I haven't really checked this but it should work I think.

[bigpooch]

If X is fixed and all the sides of the polygon are of this
length, just look at the right triangle formed by the
perpendicular to one of the sides (towards the center of
the circle) and one half of this side of the polygon (and
the other side has length equal to R, the radius of the
circle). By trig,

sin(pi/N) = (X/2)/R or
R = X/[2 sin (pi/N)].

Note sin(pi/N) approaches pi/N as N goes to infinity so the
last line makes sense intuitively.