Formula for polygon dimension

Question

In this example, if all sides are 10mm long and I did not know that the distance between the parallel lines was 17.321 what formula could be used to get that number?

Answer 1

For regular polygons with constant side length $s$ and with $n \ge 3$, if you want the length of the line segment between alternate vertices - that is, skipping one vertex in the middle (which seems to be what you're asking), the formula is $L = 2s \cos \frac{\pi} n$, where the angle is specified in radian measure. To change to degrees, just replace the $\pi$ with $180^{\circ}$.

For the case $n=6$ (a regular hexagon) like in your figure, you'll get $2(10)\cos \frac{\pi} 6 = 10\sqrt 3 \approx 17.3205$.

Proving the formula is simple. The internal angle of a regular polygon can easily be shown to be $\frac{(n-2)\pi} {n} $ by any number of elementary methods such as triangular decomposition or computing the external angles first. If you bisect this angle you get a right triangle with hypotenuse $s$, giving you half the required length you want to find using simple right triangle trigonometry as $s \sin \frac{(n-2)\pi} {2n} $, which can be simplified by a trigonometric identity to $s \cos \frac{\pi} n$. Then the required length $L$ is simply double this, i.e. $L= 2s \cos \frac{\pi} n$.