🔷 Polygons & Tilings

Polygons are fundamental geometric shapes that appear frequently in AMC problems. Master their properties and relationships for contest success.

🎯 Key Concepts

Polygon Basics

Definition: Closed figure formed by line segments Types:

  • Convex: All interior angles less than $180°$
  • Concave: At least one interior angle greater than $180°$
  • Regular: All sides and angles equal

Interior Angle Sum

Formula: Sum of interior angles = $(n-2) \cdot 180°$ where $n$ is number of sides

Individual Angles:

  • Regular polygon: Each interior angle = $\frac{(n-2) \cdot 180°}{n}$
  • Irregular polygon: Use sum formula and given information

Exterior Angle Sum

Fundamental: Sum of exterior angles = $360°$ (for any polygon)

Individual Angles:

  • Regular polygon: Each exterior angle = $\frac{360°}{n}$
  • Irregular polygon: Use sum formula and given information

Regular Polygons

Properties:

  • All sides equal
  • All angles equal
  • Can be inscribed in circle
  • Can have circle inscribed in them

Common Regular Polygons:

  • Triangle (3): Interior angle $60°$, exterior angle $120°$
  • Square (4): Interior angle $90°$, exterior angle $90°$
  • Pentagon (5): Interior angle $108°$, exterior angle $72°$
  • Hexagon (6): Interior angle $120°$, exterior angle $60°$

🔍 Micro-Examples

Interior Angle Sum

Pentagon (5 sides):

  • Sum of interior angles = $(5-2) \cdot 180° = 3 \cdot 180° = 540°$
  • Each interior angle (regular) = $\frac{540°}{5} = 108°$

Exterior Angle Sum

Any polygon:

  • Sum of exterior angles = $360°$
  • Each exterior angle (regular hexagon) = $\frac{360°}{6} = 60°$

Regular Polygon Example

Regular octagon (8 sides):

  • Each interior angle = $\frac{(8-2) \cdot 180°}{8} = \frac{6 \cdot 180°}{8} = 135°$
  • Each exterior angle = $\frac{360°}{8} = 45°$

⚠️ Common Traps

Pitfall: Wrong interior angle formula

  • Fix: $(n-2) \cdot 180°$, not $n \cdot 180°$

Pitfall: Confusing interior and exterior angles

  • Fix: Interior angles are inside, exterior angles are outside

Pitfall: Forgetting about regular polygons

  • Fix: Regular polygons have equal sides and angles

Pitfall: Wrong exterior angle sum

  • Fix: Sum of exterior angles is always $360°$

🎯 AMC-Style Worked Example

Problem: A regular polygon has interior angles measuring $150°$ each. How many sides does the polygon have?

Solution: Using the formula for interior angles of regular polygon: $\frac{(n-2) \cdot 180°}{n} = 150°$

Cross-multiplying: $(n-2) \cdot 180° = 150° \cdot n$ $180n - 360 = 150n$ $180n - 150n = 360$ $30n = 360$ $n = 12$

Answer: The polygon has 12 sides (dodecagon)

🔗 Related Topics

💡 Quick Reference

Angle Formulas

  • Interior sum: $(n-2) \cdot 180°$
  • Exterior sum: $360°$
  • Regular interior: $\frac{(n-2) \cdot 180°}{n}$
  • Regular exterior: $\frac{360°}{n}$

Common Regular Polygons

  • Triangle (3): Interior $60°$, exterior $120°$
  • Square (4): Interior $90°$, exterior $90°$
  • Pentagon (5): Interior $108°$, exterior $72°$
  • Hexagon (6): Interior $120°$, exterior $60°$
  • Octagon (8): Interior $135°$, exterior $45°$

Special Properties

  • Regular polygons: Can be inscribed in circle
  • Convex polygons: All interior angles less than $180°$
  • Concave polygons: At least one interior angle greater than $180°$

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