Explanation
In this page, we look at more examples using the exterior-angle sum of convex polygons.
The main idea is:
When we move around a convex polygon, the exterior angles make one complete turn.
One complete turn is:
\(360^\circ\)
So, no matter how many sides the convex polygon has, the sum of its exterior angles is always:
\(360^\circ\)
Formula / Rule
For any convex polygon:
Sum of Exterior Angles \(= 360^\circ\)
Also:
Exterior Angle \(= 180^\circ −\) Interior Angle
Example 1
Find the sum of the exterior angles of a convex triangle.
A convex triangle has \(3\) exterior angles.
For any convex polygon:
Sum of Exterior Angles \(= 360^\circ\)
So, for a convex triangle:
\(\text{Angle } 1 + \text{Angle } 2 + \text{Angle } 3 = 360^\circ\)
Therefore, the sum of the exterior angles is:
\(360^\circ\)
Example 2
Find the sum of the exterior angles of a convex quadrilateral.
A convex quadrilateral has \(4\) exterior angles.
For any convex polygon:
Sum of Exterior Angles \(= 360^\circ\)
So:
\(\text{Angle } 1 + \text{Angle } 2 + \text{Angle } 3 + \text{Angle } 4 = 360^\circ\)
Therefore, the sum of the exterior angles is:
\(360^\circ\)
Example 3
Find the sum of the exterior angles of a convex pentagon.
A convex pentagon has \(5\) exterior angles.
For any convex polygon:
Sum of Exterior Angles \(= 360^\circ\)
So:
\(\text{Angle } 1 + \text{Angle } 2 + \text{Angle } 3 + \text{Angle } 4 + \text{Angle } 5 = 360^\circ\)
Therefore, the sum of the exterior angles is:
\(360^\circ\)
Example 4
Find the missing exterior angle of a convex quadrilateral.
The exterior angles are:
\(70^\circ, 80^\circ, 100^\circ,\) and \(x\)
For any convex polygon:
Sum of Exterior Angles \(= 360^\circ\)
So:
\(70^\circ + 80^\circ + 100^\circ + x = 360^\circ\)
Add the known exterior angles:
\(70^\circ + 80^\circ + 100^\circ = 250^\circ\)
Now subtract from \(360^\circ\):
\(x = 360^\circ − 250^\circ\)
\(x = 110^\circ\)
So, the missing exterior angle is:
\(110^\circ\)
Example 5
Find the missing exterior angle of a convex pentagon.
The exterior angles are:
\(60^\circ, 75^\circ, 80^\circ, 65^\circ,\) and \(x\)
For any convex polygon:
Sum of Exterior Angles \(= 360^\circ\)
So:
\(60^\circ + 75^\circ + 80^\circ + 65^\circ + x = 360^\circ\)
Add the known exterior angles:
\(60^\circ + 75^\circ + 80^\circ + 65^\circ = 280^\circ\)
Now subtract from \(360^\circ\):
\(x = 360^\circ − 280^\circ\)
\(x = 80^\circ\)
So, the missing exterior angle is:
\(80^\circ\)
Practice
- More Examples (Current page)
Continue Learning
- Sum of Exterior Angles of Regular Polygons (Proof)
- Sum of Exterior Angles of Convex Polygons (Proof)