Interior and Exterior Angles of Polygons
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn how to find the sum of interior angles and exterior angles for any polygon and how to use those formulas to solve for unknown angles.
The Concept
Section titled “The Concept”A polygon is a closed shape with straight sides.
Interior Angles (angles inside the polygon):
The sum of the interior angles of any polygon with n sides is:
- Triangle (n = 3): (3 − 2) × 180° = 180°
- Quadrilateral (n = 4): (4 − 2) × 180° = 360°
- Pentagon (n = 5): (5 − 2) × 180° = 540°
- Hexagon (n = 6): (6 − 2) × 180° = 720°
Exterior Angles (angles formed by one side and the extension of an adjacent side):
The sum of the exterior angles of any polygon (one at each vertex) is always 360°, no matter how many sides it has.
Each exterior angle of a regular polygon (all sides and angles equal) is:
Worked Example
Section titled “Worked Example”-
What is the sum of the interior angles of a pentagon?
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A regular hexagon has 6 sides. What is the measure of each interior angle?
Sum = (6 − 2) × 180° = 720°. Each interior angle = 720° ÷ 6 = 120°.
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What is the measure of each exterior angle of a regular octagon (8 sides)?
Real-World Application
Section titled “Real-World Application”These formulas are very useful in real life:
- Architecture and construction: Calculating angles for roof designs, floor tiles, or polygonal rooms.
- Landscaping and gardening: Planning flower beds or patios that are pentagons, hexagons, or octagons.
- Manufacturing: Designing signs, tables, or decorative frames with multiple sides.
- Navigation and maps: Understanding angles when working with polygonal boundaries or routes.
For example, if you’re building a hexagonal gazebo, knowing each interior angle is 120° helps you cut the wood accurately.