Lesson 6.3: Regular Polygons & Interior/Exterior Angles

Learning Objectives

• Identify regular polygons and their properties.
• Calculate interior and exterior angles of regular polygons.
• Apply formulas to solve polygon problems.

Topics Covered

• Definition of regular polygons
• Interior angle formula: \((n-2) \times 180^\circ / n\)
• Exterior angle formula: \(360^\circ / n\)
• Relationship between interior and exterior angles
• Applications and problem solving with polygons

Key Terms

• Regular Polygon: Polygon with all sides and angles equal.
• Interior Angle: Angle inside a polygon at a vertex.
• Exterior Angle: Angle formed outside the polygon, supplementary to the interior angle.
• n-gon: Polygon with \(n\) sides.

Lesson Examples

Example 1: Interior Angle of a Hexagon

1. Number of sides \(n = 6\)
2. Use formula: \((n-2) \times 180^\circ / n = (6-2) \times 180^\circ / 6 = 120^\circ\)
3. Each interior angle = 120°

Example 2: Exterior Angle of a Regular Pentagon

1. Number of sides \(n = 5\)
2. Use formula: \(360^\circ / n = 360^\circ / 5 = 72^\circ\)
3. Each exterior angle = 72°

Example 3: Relationship Between Interior and Exterior Angles

1. Interior angle = 120° (hexagon)
2. Exterior angle = 180° - 120° = 60°
3. Sum of interior and exterior angles at each vertex = 180°

Practice Problems

1. Find the interior and exterior angles of a regular octagon.
2. Determine the number of sides of a regular polygon if each interior angle is 140°.
3. Calculate the sum of interior angles of a 12-sided polygon.

Tips & Common Mistakes

• Always distinguish between interior and exterior angles.
• Double-check formulas: interior = \((n-2)180/n\), exterior = 360/n.
• Sum of interior angles = (n-2) × 180°, not the measure of one angle.

Summary

Regular polygons have equal sides and angles. Interior and exterior angles can be calculated using formulas based on the number of sides. These relationships are critical for geometric problem solving.

Challenge Problems

1. Each interior angle of a regular polygon is 150°. How many sides does it have?
2. Find the exterior angles of a regular 20-sided polygon.
3. Explain why the sum of exterior angles of any polygon is always 360°.