# Can circumscribing a circle around a polygon prove that the sum of the interior angles of an n-sided polygon is $180(n - 2)$?

I am trying to create my own proof that the sum of the interior angles in a regular polygon is $180(n - 2)$, where $n$ is the number of sides in the polygon. I have seen these proofs for this formula, and I have also seen this inductive proof for the formula. I'm trying to prove this by circumscribing a circle around a polygon. My question about this is, is this a possible way to prove that the sum of the interior angles of an $n$-gon is $180(n - 2)$? If it is, I have the "first" step completed, which is to circumscribe a circle around a polygon, but I don't know where to go from here. Can anyone help me with the proof if it's possible? Here is an illustration of what I have got so far:

## Notes

I have seen this question about proving this, but this deals with an inductive proof and not a geometric proof.

• What kinds of polygon? Most polygons cannot be inscribed in a circle. – André Nicolas Jun 9 '16 at 18:50
• Any regular polygon. Will be edited. – Obinna Nwakwue Jun 9 '16 at 18:53
• That should be any regular convex polygon, to rule out regular stars. – David K Jun 9 '16 at 18:55
• It seems like your approach will end up being a variation of the Method 5 (spider theorem) at your link, or at least using many of the same ideas. – Joffan Jun 9 '16 at 18:56
• For a convex $n$-gon take an interior point $P$ and join it to each vertex. The $n$ triangles have total angles of $180n$ deg. $360$ deg is accounted for by the angles at $P$, leaving $180(n-2)$ for the angles of the $n$-gon. – almagest Jun 9 '16 at 18:59

Show that the sum of the angles in a triangle is $180^{\circ}$ (use the fact that an angle inscribed in a circle subtends twice the arc length).
Then, break up an inscribed $n$-gon ($n > 3$) into $n-2$ triangles by picking a vertex and drawing a line segment to each non-adjacent vertex.
The angles of each triangle add up to $180^{\circ}$, and also sum precisely to the angles in the $n-$gon.
• I like your intution, but will circumscribing a circle around an $n$-gon help with my proof? – Obinna Nwakwue Jun 9 '16 at 21:22