Sum of Exterior Angles of a Convex Polygon
These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 9 Mathematics Textbook, Unit 6, Chapter 6.2.
~5 min read- An exterior angle is formed outside the polygon by one side and the extension of the side next to it.
- At each vertex, the interior angle and the exterior angle add up to 180°.
- The sum of the exterior angles of any convex polygon, one at each vertex, is always 360°.
- This sum does not depend on the number of sides.
Key Words
- Exterior angle: an angle outside a convex polygon formed by one of its sides and the extension of an adjacent side.
- Adjacent side: the side next to the one you are working with. The two sides share a vertex.
- Extension of a side: the line you get when you continue a side straight past its vertex.
- Straight angle: an angle of exactly 180°, the angle of a straight line.
What Is an Exterior Angle?
- Take any side of a convex polygon and continue it straight past the vertex. The angle between this extension and the next side is an exterior angle.
- The exterior angle sits outside the polygon, right next to the interior angle at the same vertex.
- Together, the interior angle and the exterior angle make a straight line. This means interior angle + exterior angle = 180° at every vertex.
- Be careful: not every angle drawn outside a polygon is an exterior angle. The angle must be formed by one side and the extension of the side next to it.
- Each vertex of a polygon has one exterior angle on each side of it, but when we add exterior angles we count only one per vertex.
The Sum Is Always 360°
- Here is the key result of this chapter: for any convex polygon, the exterior angles, one at each vertex, always add up to 360°.
- The proof is short. At each of the n vertices, the interior and exterior angles make a straight line, so all the pairs together make n × 180°.
- The interior angles alone add up to (n − 2) × 180°, which equals n × 180° − 360°.
- Subtract: the exterior angles add up to n × 180° − (n × 180° − 360°) = 360°.
- Notice that n cancels out completely. A triangle, a hexagon and a 100-gon all have the same exterior angle sum of 360°.
Worked example: show that the exterior angles of a triangle add up to 360°.
A triangle has 3 vertices, and each interior-exterior pair makes 180°, so all pairs together make 3 × 180° = 540°.
The interior angles of a triangle add up to 180°.
So the exterior angles add up to 540° − 180° = 360°.
Worked example: two interior angles of a triangle are 60° and 40°. Find the exterior angle at each vertex.
First find the third interior angle: x + 60° + 40° = 180°, so x = 80°.
Each exterior angle is 180° minus the interior angle at that vertex.
The exterior angles are 180° − 60° = 120°, 180° − 40° = 140° and 180° − 80° = 100°.
Check the sum: 120° + 140° + 100° = 360°. The rule works.
Common Mistakes to Avoid
- Computing the exterior angle as 360° − x instead of 180° − x, where x is the interior angle. The interior and exterior angles lie on a straight line, so they add up to 180°, not 360°.
- Thinking the exterior angle sum grows when the polygon gets more sides. The interior sum grows, but the exterior sum stays fixed at 360°.
- Treating any angle outside the polygon as an exterior angle. The angle between two extended sides, for example, does not count because it is not formed by a side and the extension of its adjacent side.
- Counting two exterior angles at the same vertex when adding. The 360° rule uses exactly one exterior angle per vertex.
Easy Ways to Remember
- Imagine walking around the edge of a polygon-shaped field. At each corner you turn through the exterior angle, and when you arrive back at the start you have made one full turn of 360°.
- Interior sums change with n, but the exterior sum is always 360°. One full circle covers every polygon.
- At every vertex, think “straight line”: interior + exterior = 180°.
- To find an exterior angle quickly, subtract the interior angle from 180°.
Quiz
Tap an answer to check it.
1. What is the sum of the exterior angles of a convex hexagon, one at each vertex?
The exterior angles of any convex polygon add up to 360°, no matter how many sides it has. The value 720° is the sum of the interior angles of a hexagon.
2. At one vertex of a convex polygon, the interior angle is 120°. What is the exterior angle at that vertex?
The interior and exterior angles at a vertex lie on a straight line, so they add up to 180°. The exterior angle is 180° − 120° = 60°. The answer 240° comes from wrongly subtracting from 360°.
3. The exterior angles of a pentagon are (n + 5)°, (2n + 3)°, (3n + 2)°, (4n + 1)° and (5n + 4)°. What is the value of n?
The five exterior angles add up to 360°. Adding the expressions gives 15n + 15 = 360, so 15n = 345 and n = 23.
4. The interior angles of a triangle are 50°, 60° and 70°. What are its exterior angles?
Subtract each interior angle from 180°: 180 − 50 = 130, 180 − 60 = 120 and 180 − 70 = 110. Check: 130° + 120° + 110° = 360°.
5. Which one of the following statements is correct for every convex polygon?
The interior and exterior angles at a vertex always make a straight line of 180°. The exterior angle sum is fixed at 360° for every convex polygon, and each exterior angle of a convex polygon is less than 180°.
Remember: The exterior angles of any convex polygon add up to 360°, one at each vertex, no matter how many sides the polygon has. At every vertex, the interior and exterior angles make a straight line: they add up to 180°.