The formula for the sum of interior angles of any polygon — built directly from the triangle's 180° rule.
The Mnemonic
Every Polygon Is Made of Triangles
The sum of a polygon's interior angles depends only on how many sides it has, following the formula: Sum = (n − 2) × 180°, where n is the number of sides.
For a regular polygon (all sides and angles equal), dividing this sum by n gives the measure of just ONE interior angle: Each angle = [(n − 2) × 180°] / n.
Drawing diagonals from one vertex splits any polygon into triangles — a pentagon (5 sides) always makes exactly 3
💡 Memory Trick
"Formula for the sum of interior angles of any polygon." (n-2) × 180° — the "n-2" counts how many triangles you can split the polygon into by drawing diagonals from a single vertex, and each triangle contributes exactly 180° (straight from the Triangle Angle Sum Theorem).
Why It Works
Where the (n-2) Comes From
Pick any single vertex of a polygon and draw diagonals from it to every other non-adjacent vertex. This splits the polygon into a series of triangles, all sharing that one starting vertex. For a triangle (n=3), you get 1 triangle (no diagonals needed). For a quadrilateral (n=4), you get 2 triangles. For a pentagon (n=5), you get 3 triangles — the count is always exactly n−2.
Since every triangle's angles sum to 180° (Triangle Angle Sum Theorem), and the polygon's full interior angle sum is just all of those triangles' angles combined, the polygon's total angle sum is (n−2) triangles × 180° each = (n−2) × 180°. This formula is really just the Triangle Angle Sum Theorem applied repeatedly.
Using It In A Proof
Total Sum vs. One Angle
The most important distinction in this topic is whether a problem wants the TOTAL sum of all interior angles, or the measure of just ONE angle in a regular polygon — these require different final steps.
1
Count the sides (n) carefully
Count the actual number of sides of the polygon — this single number drives the entire formula.
2
For total sum, stop after multiplying
Sum = (n−2) × 180° — this is the final answer if the question asks for the TOTAL interior angle sum.
3
For one angle in a regular polygon, divide by n
If the polygon is regular AND the question asks for one angle's measure, take the total sum and divide by n.
Full Worked Example
Finding One Angle of a Regular Polygon
Given: A regular octagon (8 sides). Find: the total interior angle sum, and the measure of each individual angle.
Since the octagon is regular, each individual angle = 1080° / 8.
3
Calculate
Each interior angle = 135°.
This division step (÷n) only applies to REGULAR polygons — an irregular octagon still has a total angle sum of 1080°, but its individual angles could be any values that add up to that total.
🎯 Quick Worked Example
A polygon has an interior angle sum of 900°. How many sides does it have?
1
Set up the equation. (n−2) × 180 = 900.
2
Solve for n. n−2 = 900/180 = 5, so n = 7.
3
Conclude. The polygon has 7 sides (a heptagon).
📌 Exam Application
Working backward from a given angle sum to find n (as in the worked example above) is just as commonly tested as the forward direction — recognize that the formula can be solved for n by dividing the given sum by 180° and adding 2, rather than only ever plugging n in to find a sum.
⚠️ Most Common Polygon Angle Sum Mistakes
Trap 1 — Forgetting to subtract 2 from n: Using n × 180° instead of (n−2) × 180° is a common slip that overstates the angle sum — always subtract 2 before multiplying.
Trap 2 — Dividing by n when the polygon isn't regular: Dividing the total sum by n to get 'one angle' only works for regular polygons, where every angle is equal — for an irregular polygon, individual angles can vary even though they still add up to (n−2)×180° in total.
✓ Quick Self-Test
1) State the Polygon Angle Sum formula. 2) Why does the formula subtract 2 from n? 3) Find the interior angle sum of a hexagon (6 sides). 4) A regular decagon has 10 sides — find the measure of one interior angle. 5) A polygon has an interior angle sum of 1440° — how many sides does it have?