Cierra Castillo

2022-07-13

I'm not great at mathematics so I'm sure this is trivial to most. I have been searching around however and not been able to find how to figure out the incircle of a circle sector, or, in other words, the point inside a sector that is furthest away from the radii and the arc. The simpler solution the better

Charlee Gentry

Expert

Assume the sector angle is $2\theta$ and the radius is 1, with center at (0,0) and one end of the arc at (1,0). Then the angle bisector of the sector, on which the center $O$ of the incircle must lie, is at angle $\theta$ from the $x$ axis. Then $O=\left(r\mathrm{cos}\theta ,r\mathrm{sin}\theta \right)$ where $r\mathrm{sin}\theta =1-r.$ Then $r$ may be found from this. Of course the whole diagram may have to be rescaled and rotated depending on how your sector is situated.