Demonstrate by mathematical induction that the sum of the internal angles of a $n$-sided concave polygon is $180^\circ(n-2)$.

Have I solved this classic problem for convex polygons, any tips on how to connect the two solutions?

  • 3
    $\begingroup$ They say "demonstrate by induction". To me, that hints towards cutting off a triangle somewhere. Or just cut anywhere between two vertices that can "see" one another. $\endgroup$
    – Arthur
    Oct 8 '19 at 18:24

We can restate the claim as this: $\pi$ times the number of sides, minus the sum of internal angles, is $2\pi$.

Suppose a reflex angle occurs at a vertex $B$ from $A$ to $C$. If you replace the polygonal arc $ABC$ with $AC$, you reduce the number of reflex angles by $1$, while reducing the sum of internal angles by $\pi$ radians, because if the reflex angle was $\pi+x$ we replace it with two contributions totalling $\pi-(\pi-x)=x$. We've also reduced the number of sides by $1$, preserving the above quantity we claim to be invariant.

So instead of inducting on the number of sides, induct on the number of reflex angles. Your base case, where there are none of them, is the convex polygons you already understand.

  • $\begingroup$ I'm not convinced the number of reflex angles always decreases: I think angles at $A$ and $C$ in your example could become reflex angles after replacing $ABC$ with $AC$. $\endgroup$ Oct 9 '19 at 11:36
  • $\begingroup$ @Aretino It won't always reduce the number of reflex angles, but it does always reduce the number of vertices, so if you keep doing it you'll eventually have so few of them no angles can be reflex. If you then inspect what you've done backwards, you'll see any planar polygon can be obtained by gradually adding reflex angles to a convex polygon. $\endgroup$
    – J.G.
    Oct 9 '19 at 11:59

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