# Distance between center of side of regular polygon inscribed in a circle, and the perimeter of that circle?

Point A : The center of a side of a polygon inscribed in a circle

Point B : The point on the perimeter of that circle that is opposite Point A

I want to calculate the distance between Point A & Point B

I know the number of sides the polygon has and the radius of the circle it's inscribed in.

I believe this is the way to calculate the length of the side of a regular polygon with n sides (Please correct me if I'm wrong as I am not a mathematician and found this online) :

x = nrsin(π/n)


Now is the distance I'm looking for : distance = 2r - x ?

I'm trying to figure out a formula that I could use in a bit of code I'm writing

• Do you mean an inscribed polygon? So that the vertices are on the circle itself? In that case the distance is 0 because the side is touching the circle. If you want to calculate the distance from the center of a side, draw a triangle with vertices at the center of the circle and at the ends of a side and subtract the altitude from the radius. – Matt Samuel Nov 1 '14 at 19:51
• Ah yes, sorry for not being clear Matt. I want to calculate the distance between the center of a side of the polygon and the point on the perimeter of the circle right opposite that center. I want to use Math to do that as I will use it a bit of Javascript code I am writing. I will update the OP. – Sprout Coder Nov 1 '14 at 19:56

The length of a side is $2r sin(\frac{\pi}{n})$ by trigonometry.
The distance is $r(1-cos\frac{\pi}{n})$, which is the difference between the radius and the length of the line from the center of the circle to the center of the side.