Sum of inner angles of polygon

Summary

For an $n$-sided polygon, the sum of the inner angles is:

$$\sum _{i=1}^n{\theta }_i^{\text{inner}}=180(n-2)$$

If $n=3$, then we see that the sum of inner angles if $180(3-2)=180$, which is a fact I memorized as a student. We first prove this using a triangle, and then discuss how our proof generalizes to all $n$-sided polygons.

Proof

First, consider a triangle described by the points $A,B,C$, with respective inner angles ${\theta }_1^{\text{inner}},{\theta }_2^{\text{inner}},{\theta }_3^{\text{inner}}$. Note that these angles do not necessarily equal each other.

We have also written out the turn angles, ${\theta }_1^{\text{turn}},{\theta }_2^{\text{turn}},{\theta }_3^{\text{turn}}$, by extending out the lines of the triangle. The turn and inner angles are supplementary, meaning each pair of inner/turn angles sums to $180$:

$$\begin{array}{rlrlrlr}{\theta }_i^{\text{inner}}+{\theta }_i^{\text{turn}}& =180& & \forall i\in \{1,2,3\}& & & \text{(Def. of supplementary angles)}\\ ⟹{\theta }_i^{\text{turn}}& =180-{\theta }_i^{\text{inner}}& & \text{(Rearrange equation)}& & & \end{array}$$

Additionally, we know the sum of the turn angles equals $360°$. We will prove this by considering a person walking the entire perimeter of the shape counter-clockwise (i.e., always heading in the direction of the arrows on the triangle edges), and turning ${\theta }_i^{\text{turn}}$ degrees at each point:

• First, the person standing at point $A$, and facing towards point $B$, and the total degrees they have turned is $0$.
• They will walk towards $B$, and then turn towards point $C$ by turning counter-clockwise ${\theta }_2^{\text{turn}}$ degrees. Therefore, the total degrees turned is now ${\theta }_2^{\text{turn}}$.
• They will next walk towards $C$, and then turn towards point $A$ by turning counter-clockwise ${\theta }_3^{\text{turn}}$ degrees. Therefore, the total degrees turned is now ${\theta }_2^{\text{turn}}+{\theta }_3^{\text{turn}}$.
• They will last walk towards $A$, and then turn towards point $B$ by turning counter-clockwise ${\theta }_1^{\text{turn}}$ degrees. The person has now walked the entire perimeter and turned a total degrees of ${\theta }_2^{\text{turn}}+{\theta }_3^{\text{turn}}+{\theta }_1^{\text{turn}}$. Not only is the person back at the starting point after all three turn turns, but they are also facing the same direction they started. Because each turn was counter-clockwise (so each turn positively contributes to the total turn) and that the shape is a polygon (so no lines cross-over), ending up facing the same orientation as we started means the person spun a total of $360°$. Therefore, we conclude that:

$$\begin{array}{rlrl}{\theta }_1^{\text{turn}}+{\theta }_2^{\text{turn}}+{\theta }_3^{\text{turn}}& =360& & \text{(Constraint on turn angles)}\end{array}$$

We can now substitute in our rearranged version of the definition of supplementary angles into the constraint on turn angles equation. Here, I will use $n=3$ to represent the number of sides/inner angles of a triangle:

$$\begin{array}{rlrl}\sum _{i=1}^n{\theta }_i^{\text{turn}}& =360& & \text{turn angle constraint equation)}\\ \sum _{i=1}^n(180-{\theta }_i^{\text{inner}})& =360& & \text{(Substitution)}\\ 180n+\sum _{i=1}^n(-{\theta }_i^{\text{inner}})& =360& & \text{(Factor out 180)}\\ \sum _{i=1}^n{\theta }_i^{\text{inner}}& =-360+180n& & \text{(Rearrange)}\\ \sum _{i=1}^n{\theta }_i^{\text{inner}}& =180(n-2)& & \text{(Factor out 180)}\end{array}$$

This is exactly the formula we know and love for the sum of inner angles of a $n$-sided polygon, which gives us the answer for a triangle: $180°$.

You’ll notice that our proof above did not rely on anything about the fact that it was a triangle: the triangle just meant that the polygon had $3$ sides/angles. Therefore, we can apply our proof to all $n$-sided polygon, which is why we wrote the above derivation with $n$.

Credit

Thank you to Michael Fishman for providing the proof and inspiration to write this note, Zoheb Anjum for additional clarity, and Thao Nguyen for reviewing/make suggestions to this note.