The Polygon Factory
Syllabus 4.2: Interior & Exterior Angles • Regular Polygons • The $360°$ Rule
The Polygon Factory
The Explosion Rule
All exterior angles collapse into a perfect circle: 360°
The Logic Track
Phase 1: The Exterior Hack
Always find the Exterior Angle first! It's easier: just divide 360° by the number of sides. For a regular 5-gon, each exterior angle is 72.0°.
Exterior Angle = 360° ÷ 5 = 72.0°
Phase 2: The Straight Line Rule
Interior and Exterior angles are supplementary(they add to 180°). So subtract the exterior angle from 180° to find the interior angle.
Interior Angle = 180° − 72.0° = 108.0°
Phase 3: The Triangle Rule
A polygon is just triangles glued together. An 5-gon splits into 3 triangles. Each triangle contributes 180°, giving a total of 540°.
Sum = (5 − 2) × 180° = 3 × 180° = 540°
Examiner Trap Alert!
Common Mistake: Students forget that the formula (n − 2) × 180° gives the sum of all interior angles. To find one interior angle in a regular polygon, you must divide by n!
One Interior Angle = [(n − 2) × 180°] ÷ n
Dynamic Polygon Calculator
Problem: Find the interior angle and sum of interior angles for a regular 5-gon.
Find Exterior Angle
Exterior Angle = 360° ÷ 5
= 72.00°
Find Interior Angle
Interior Angle = 180° − Exterior Angle
= 180° − 72.00°
= 108.00°
Find Sum of Interior Angles
Sum = (n − 2) × 180°
= (5 − 2) × 180°
= 3 × 180°
= 540°
Verification (Alternative Method)
Sum = One Interior Angle × Number of Sides
= 108.00° × 5
= 540.00° ✓
Answer: Each interior angle = 108.00° | Sum of interior angles = 540°
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