Polygon Exterior Angle Sum
No matter how many sides a convex polygon has, its exterior angles always sum to exactly . It's one of geometry's most elegant surprises.
Introduction
Imagine walking along the edges of a polygon — turning at each vertex. When you return to where you started, you've made exactly one full rotation: 360°. Each “turn” is an exterior angle.
Regular pentagon: each exterior angle = 72°
Past Knowledge
Interior angle sum (9.1.2). Linear pairs (1.3.4). Supplementary angles.
Today's Goal
Prove and apply the Exterior Angle Sum Theorem for polygons.
Future Success
Quadrilateral properties (9.2), tessellations, regular polygon constructions.
Key Concepts
Exterior Angle Sum Theorem
For any convex polygon, one exterior angle at each vertex:
This is true regardless of the number of sides!
Each Exterior Angle of a Regular Polygon
Theorem & Proof
Two-Column Proof: Exterior Angle Sum = 360°
Given: A convex -gon with one exterior angle at each vertex
Prove: Sum of all exterior angles
| # | Statement | Reason |
|---|---|---|
| 1 | At each vertex, interior + exterior = 180° | Linear pair postulate |
| 2 | Sum of all pairs: | vertices, each contributing 180° |
| 3 | Total = (sum of interiors) + (sum of exteriors) | Partitioning the |
| 4 | Interior sum from Lesson 9.1.2; let = exterior sum | |
| 5 | Algebra — subtract interior sum |
∎ The cancels perfectly, leaving exactly 360° every time — independent of .
Worked Examples
Each Exterior Angle
Find each exterior angle of a regular 12-gon.
30°
Find n from Exterior Angle
A regular polygon has each exterior angle = 24°. How many sides?
15-gon
Missing Exterior Angle
A hexagon has five exterior angles: 55°, 72°, 38°, 65°, 80°. Find the sixth.
50°
Common Pitfalls
Using (n−2)·180° for Exterior Angles
That formula is for interior angles. Exterior angles always sum to 360° — period. No in the sum formula.
Picking Two Exterior Angles at One Vertex
Each vertex has TWO exterior angles (they're vertical angles). The theorem uses one at each vertex. Don't double-count.
Real-Life Applications
Logo Design — Spirograph Patterns
Designers who create circular star and polygon patterns use 360° ÷ n to space the points evenly. Every company logo with a circular star pattern uses this principle.
Robotics — Turning a Corner
A robot navigating a polygonal room needs to turn through each exterior angle. Programming it to check that the total is 360° verifies it completed the loop.
Practice Quiz
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