The sum of оbtuse angles of а convex polygon is $2013^\circ$. How many sides does it have?

Question

The sum of оbtuse angles of а convex polygon is $2013^\circ$. How many sides does it have?

I haven't seen a fоrmula that express this kind of relationship. Can you give me some points?

Answer 1

The sum of the exterior angles of a convex polygon is $360^\circ$. Since each acute angle contributes more than $90^\circ$ to this count, there can’t be more than $3$ of them. This implies that the total sum of the angles of the polygon is between $2013^\circ$ and $2283^\circ$.

However, the sum of the angles of a convex polygon is always a multiple of $180^\circ$ – more specifically, it’s given by $$S=(n-2)180^\circ,$$ where $n$ is the number of sides on the polygon. The only way this is possible is if $S=2160^\circ$ and $n=\boxed{14}$. $\blacksquare$