# Thread: How to calculate the distance from center to sides.

1. ## How to calculate the distance from center to sides.

Hi smart people.

Sorry, i might not even be able to formulate my question correct, so i made a sketch
I need to find the distance from the center of the shape, to the center of the sides. See JPG.

Known variable values are the width of the sides (in this example 100), and amount of sides.

So for an example, if i need to have 10 sides, what would the distance be? (green line)

https://imgur.com/a/eyU5W (if you cant see above pic)

Thanks

Edit: The green values on the sketch, are found simply by measuring

2. ## Re: How to calculate the distance from center to sides.

Since you appear to only be dealing with regular polygons, the sum of the internal angles will be (in radians) $(n-2)\pi$ where $n$ is the number of sides. If you were to draw lines from the center to two adjacent corners, you would wind up with an isosceles triangle. One angle would be $\dfrac{n-2}{n}\pi$ while the other two would be $\dfrac{\pi- \dfrac{n-2}{n}\pi}{2} = \dfrac{\pi}{n}$ radians. If each side has a length of $k$ and the distance you are looking for has a length of $x$, then we can apply the Law of Sines:

$\dfrac{\sin \left( \dfrac{n-2}{2n}\pi \right) }{\left( \dfrac{k}{2} \right) } = \dfrac{\sin \left( \dfrac{\pi}{n} \right) }{x}$

$x = \dfrac{k\sin \left( \dfrac{\pi}{n} \right) }{ 2 \sin \left(\dfrac{n-2}{2n} \pi \right) }$

In your example, you are looking for $x$ where $k = 100, n=10$.

$x = \dfrac{50 \sin \left( \dfrac{\pi}{10} \right)}{ \sin \left( \dfrac{2\pi}{5} \right) } = 10\sqrt{25 - 10\sqrt{5}} \approx 16.25$

3. ## Re: How to calculate the distance from center to sides.

That is a perpendicular bisector of the side. Hence its length is of a side, . So the length of a side is $s$. So if $R$ is the radius of the circle then the red line is $\sqrt{R^2-(0.5s)^2}$