Angles

Grade 12 Math - Angles in Polygons

Lesson Objectives

Recall angle sum rules for polygons
Calculate interior and exterior angles of regular and irregular polygons
Solve geometric problems involving angle relationships in polygons

Lesson Introduction

Polygons are 2D closed figures with straight sides. Understanding how to compute interior and exterior angles is essential for solving geometric problems. We’ll explore how to apply angle formulas and visualize the concepts using diagrams.

Core Lesson Content

Polygon Angle Rules

Interior angle sum: \((n - 2) \times 180^\circ\)
Each interior angle (regular polygon): \(\frac{(n - 2) \times 180^\circ}{n}\)
Each exterior angle (regular polygon): \(\frac{360^\circ}{n}\)

Sample Polygon Diagrams

Regular Pentagon (5 sides)

Worked Examples

Example 1:
Find the sum of interior angles of a hexagon.
Solution: \((6 - 2) \times 180^\circ = 720^\circ\)

Example 2:
Find each interior angle of a regular octagon.
Solution: \(\frac{(8 - 2) \times 180^\circ}{8} = 135^\circ\)

Example 3:
Each interior angle of a regular polygon is \(150^\circ\). How many sides does it have?
Solution: Exterior angle = \(30^\circ \Rightarrow n = \frac{360^\circ}{30^\circ} = 12\)

Example 4:
Find the exterior angle of a regular decagon.
Solution: \(\frac{360^\circ}{10} = 36^\circ\)

Example 5:
Sum of the exterior angles of a 20-gon?
Solution: Always \(360^\circ\)

Example 6:
A polygon has an interior angle of \(160^\circ\). How many sides does it have?
Solution: Exterior angle = \(20^\circ \Rightarrow n = \frac{360^\circ}{20^\circ} = 18\)

Example 7:
Find the total interior angle sum of a nonagon.
Solution: \((9 - 2) \times 180^\circ = 1260^\circ\)

Example 8:
What is each interior angle of a regular 12-gon?
Solution: \(\frac{(12 - 2) \times 180^\circ}{12} = 150^\circ\)

Example 9:
If sum of interior angles is \(1980^\circ\), find number of sides.
Solution: \((n - 2) \times 180^\circ = 1980 \Rightarrow n = 13\)

Example 10:
Find the exterior angle of an 18-sided regular polygon.
Solution: \(\frac{360^\circ}{18} = 20^\circ\)

Exercises

[WAEC] Find the sum of interior angles of a dodecagon. [Past Question]
[WAEC] What is each interior angle of a regular heptagon?
If the sum of interior angles is \(2160^\circ\), how many sides?
Calculate each exterior angle of a regular 30-gon.
Find the interior angle of a regular 15-gon.
[NECO] A polygon has exterior angle \(24^\circ\). Find number of sides. [Past Question]
Confirm the sum of exterior angles for any polygon is \(360^\circ\)
[NECO] If the interior angle is \(162^\circ\), how many sides? [Past Question]
The total interior angle of a polygon is \(2340^\circ\). Find sides.
Find exterior angle of a polygon with 9 sides.

Conclusion/Recap

Polygon angle rules allow you to determine unknown angle measures quickly. These calculations are foundational in geometry, construction, and design. Always remember: interior angles depend on sides, and exterior angles always sum up to \(360^\circ\).