Calculate number of sides of cylinder so each side is a certain width

Question

I'm working on a video-game and as part of the level, I need to create one half of the room curved. For the cylinder, all sides should be of width 450cm, and the cylinder will have radius of 1475cm, because this is half the room width. This is only for one half of the room as the other half is a square. Imagine a square with half of a cylinder attached so you have one side curved. I will be using a cylinder and not a circle, so each side is flat.

How can I work out from that radius, knowing that each side should 450cm in width, the number of sides I need to put on the cylinder? So that each end of the half cylinder attaches to the edges of the square.

Answer 1

So you are approximating a circle with a polygon (the vertical size doesn't matter). Half-circle of the given radius has a length of $l=\pi r\approx 4633.9\,\rm cm$. Divide this with "side length" to get approximately how many sides you need: you get around 10.3. Of course, because you don't have a half-circle but a polygon, this is not accurate, but it doesn't matter, because you'll have to round to the nearest integer anyway. If you make it 10 sides, they will be a bit longer than 450cm, but you can't make it exact without having some leftovers.

So... you need 10 pieces, which means each side takes 180°/10=18° of the angle. At the given radius, the exact length of each side will be $2r \sin (18^\circ/2)=461.48\,\rm cm$. This is as close as you can get. With 11 sides, the sides will be 410.56cm wide.

Answer 2

If each side is $450$cm and the radius is $1475$cm, then the angle ($\theta$) at the apex of the isosceles triangle with base $450$cm and side lengths $1475$cm each is given by:$$\theta=2*\arctan(\frac{225}{1475})$$ Since the curved part of the room has to sweep $180^{\circ}$ (or $\pi$ radians) then the number of sides you need ($n$) will be given by:$$n=\frac{\pi}{\theta}=\frac{\pi}{2*\arctan(\frac{225}{1475})}\approx10.38$$