Exterior angles help describe how a polygon turns as you move around its boundary. At each vertex, an exterior angle is formed by extending one side of the polygon and measuring the angle outside the shape. These angles are useful because they reveal a simple pattern shared by all convex polygons.
No matter how many sides the polygon has, one exterior angle at each vertex always adds to 360 degrees.
The 360 degree total comes from the idea of making one complete turn as you travel around the polygon and return to your starting direction. In a regular polygon, all exterior angles are equal, so each one is found by dividing 360 degrees by the number of sides. Exterior angles are also linked to interior angles because an interior angle and its adjacent exterior angle form a straight line.
This makes exterior angles a powerful shortcut for finding missing angles and identifying regular polygons.
Key Facts
- Sum of one exterior angle at each vertex of any convex polygon = 360°.
- For a regular n-sided polygon, each exterior angle = 360°/n.
- Interior angle + adjacent exterior angle = 180°.
- For a regular n-sided polygon, each interior angle = 180° - 360°/n.
- Number of sides of a regular polygon = 360°/exterior angle.
- A convex polygon has all interior angles less than 180° and no vertices pointing inward.
Vocabulary
- Exterior angle
- An exterior angle is an angle formed outside a polygon by extending one side at a vertex.
- Interior angle
- An interior angle is an angle inside a polygon formed by two sides that meet at a vertex.
- Convex polygon
- A convex polygon is a polygon with no inward dents, so every interior angle is less than 180 degrees.
- Regular polygon
- A regular polygon is a polygon with all sides equal and all interior angles equal.
- Supplementary angles
- Supplementary angles are two angles whose measures add to 180 degrees.
Common Mistakes to Avoid
- Adding all possible exterior angles at every vertex is wrong because the 360° rule uses only one exterior angle at each vertex.
- Using 180° instead of 360° for the exterior angle sum is wrong because 180° is the sum of a linear pair, not the full turn around a polygon.
- Dividing 360° by the number of sides for an irregular polygon is wrong because that gives each exterior angle only when the polygon is regular.
- Forgetting that interior angle + exterior angle = 180° is wrong because the exterior angle is usually adjacent to the interior angle on a straight line.
Practice Questions
- 1 A regular octagon has 8 sides. What is the measure of each exterior angle?
- 2 Each exterior angle of a regular polygon measures 24°. How many sides does the polygon have?
- 3 Explain why the exterior angles of a convex polygon add to 360° even if the polygon is not regular.