How many reflex angles can a polygon have? ![Various irregular polygons higlighting reflex angles]( think the first of the polygons that can have a reflex angle is the pentagon. For a hexagon, a maximum of 2 reflex angles is possible. I tried to draw many concave polygons to find out a relation, but I got stuck. Is there a formula for this?

The maximum number of reflex angles a simply connected $n-$gon can have is $n-3$. This comes from the fact that the sum of the internal angles of a simply connected $n-$gon is $(n-2)180^\circ$ so there can be at most $n-3$ angles greater than $180^\circ$. To prove we can achieve this, take an equilateral triangle and bend one edge inward to make $n-3$ angles, all slightly greater than $180^\circ$. Below is a heptagon with four reflex angles. I think it is clear how to get as many reflex angles you want up to $n-3$

![enter image description here](