Polygon Interior Angle Sum
The sum of all interior angles of an -gon is . This formula comes from splitting the polygon into triangles.
Introduction
A triangle's angles sum to 180°. A quadrilateral's sum to 360°. What about a pentagon? A 20-gon? By drawing diagonals from one vertex, you can split any polygon into triangles and use the pattern.
Past Knowledge
Triangle Angle Sum = 180° (5.1.2). Polygon names (9.1.1).
Today's Goal
Derive and apply the interior angle sum formula for any polygon.
Future Success
Exterior angle sum (9.1.3), quadrilateral properties (9.2).
Key Concepts
Interior Angle Sum Formula
where = number of sides
Each Angle of a Regular Polygon
Quick Reference
| Polygon | Angle Sum | Regular Each | |
|---|---|---|---|
| Triangle | 3 | 180° | 60° |
| Quadrilateral | 4 | 360° | 90° |
| Pentagon | 5 | 540° | 108° |
| Hexagon | 6 | 720° | 120° |
| Octagon | 8 | 1080° | 135° |
| Decagon | 10 | 1440° | 144° |
Theorem & Proof
Two-Column Proof: Polygon Interior Angle Sum Theorem
Given: A convex polygon with vertices
Prove: Interior angle sum
Strategy: Draw all diagonals from one vertex to triangulate the polygon.
| # | Statement | Reason |
|---|---|---|
| 1 | Choose any vertex of the -gon | Arbitrary vertex selection |
| 2 | Draw diagonals from to every non-adjacent vertex | can connect to non-adjacent vertices |
| 3 | These diagonals divide the polygon into non-overlapping triangles | diagonals create regions |
| 4 | The angles of all triangles exactly cover all interior angles of the polygon | Diagonals lie inside a convex polygon; no overlap, no gaps |
| 5 | Interior angle sum | Each triangle has angle sum 180° (Triangle Angle Sum Theorem) |
∎ The triangulation approach works because every convex polygon can be divided into triangles from a single vertex.
Worked Examples
Angle Sum of a Hexagon
Find the sum of the interior angles of a hexagon.
720°
Finding a Missing Angle
A pentagon has four angles: 120°, 90°, 115°, 100°. Find the fifth.
Sum =
Fifth angle =
115°
Reverse — Find n
Each interior angle of a regular polygon is 156°. How many sides?
→
→
15-gon (pentadecagon)
Common Pitfalls
Using n Instead of (n − 2)
The formula is , NOT . You subtract 2 because the polygon splits into triangles.
Confusing Total Sum with Each Angle
The formula gives the total sum. For each angle in a regular polygon, divide by .
Real-Life Applications
Stop Signs
A regular octagon has each interior angle = . This universally recognized shape is mandated worldwide.
Tile Design
Only regular polygons with interior angles that divide evenly into 360° can tessellate alone: triangles (60°), squares (90°), and hexagons (120°).
Practice Quiz
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