You proved in Triangle Sum and Exterior Angle Proofs that a triangle's angles add to $180^\circ$. Every polygon can be cut into triangles, so the same fact gives the angle sum of any polygon.
Why it works
Interior angles. In a convex polygon with $n$ sides, draw every diagonal from one vertex. They cut it into $n - 2$ triangles (each side except the two at that vertex gets its own triangle), and the triangles' angles together make up exactly the polygon's angles. So the sum is $(n - 2) \cdot 180^\circ$.
Exterior angles. At each vertex, the interior and exterior angles form a linear pair, so all $n$ pairs total $n \cdot 180^\circ$. Subtract the interior sum: $n \cdot 180^\circ - (n - 2) \cdot 180^\circ = 360^\circ$. It is one full turn, the amount you turn while walking once around the polygon.
Exterior angles. At each vertex, the interior and exterior angles form a linear pair, so all $n$ pairs total $n \cdot 180^\circ$. Subtract the interior sum: $n \cdot 180^\circ - (n - 2) \cdot 180^\circ = 360^\circ$. It is one full turn, the amount you turn while walking once around the polygon.
| Sides $n$ | Triangles | Interior sum | Each angle if regular | Each exterior angle if regular |
|---|---|---|---|---|
| 3 | 1 | $180^\circ$ | $60^\circ$ | $120^\circ$ |
| 4 | 2 | $360^\circ$ | $90^\circ$ | $90^\circ$ |
| 5 | 3 | $540^\circ$ | $108^\circ$ | $72^\circ$ |
| 6 | 4 | $720^\circ$ | $120^\circ$ | $60^\circ$ |
| 8 | 6 | $1080^\circ$ | $135^\circ$ | $45^\circ$ |
| $n$ | $n - 2$ | $(n - 2) \cdot 180^\circ$ | $\frac{(n - 2) \cdot 180^\circ}{n}$ | $\frac{360^\circ}{n}$ |
Key idea
The interior angles of a convex $n$-gon add to $(n - 2) \cdot 180^\circ$. The exterior angles, one at each vertex, always add to $360^\circ$.
Worked example: An interior sum
Find the sum of the interior angles of a nonagon (9 sides).
- $(9 - 2) \cdot 180 = 7 \cdot 180 = 1260$.
Answer: $1260^\circ$
Worked example: Working backward from one angle
Each interior angle of a regular polygon measures $156^\circ$. How many sides does it have?
- Each exterior angle is $180 - 156 = 24^\circ$.
- The exterior angles add to $360^\circ$: $n = \frac{360}{24} = 15$.
Answer: 15 sides
Worked example: A missing angle
A hexagon has angles $110^\circ$, $135^\circ$, $120^\circ$, $140^\circ$, $95^\circ$ and $x^\circ$. Find $x$.
- Interior sum: $(6 - 2) \cdot 180 = 720$.
- Known angles: $110 + 135 + 120 + 140 + 95 = 600$.
- $x = 720 - 600 = 120$.
Answer: $x = 120$
Common mistake
Thinking the exterior angles add to more when there are more sides. The interior sum grows by $180^\circ$ with each new side, but the exterior sum stays $360^\circ$: more exterior angles, each one smaller. Check: a square has four $90^\circ$ exterior angles and a regular octagon has eight $45^\circ$ ones; both total $360^\circ$.