Explanation
In this lesson, we prove that the sum of the exterior angles of a convex polygon is always:
\(360^\circ\)
A convex polygon is a polygon where all interior angles are less than \(180^\circ\).
An exterior angle is formed outside the polygon when one side is extended.
At each vertex, the interior angle and the exterior angle form a straight angle.
A straight angle is:
\(180^\circ\)
So:
Exterior Angle \(= 180^\circ −\) Interior Angle
To understand why the exterior angles add up to \(360^\circ,\) imagine walking around the outside of a convex polygon.
Each time you reach a vertex, you turn by the exterior angle.
After walking all the way around the polygon, you have made one complete turn.
One complete turn is:
\(360^\circ\)
So, the sum of the exterior angles of a convex polygon is always:
\(360^\circ\)
This works for triangles, quadrilaterals, pentagons, hexagons, and any convex polygon.
Formula / Rule
For any convex polygon:
Sum of Exterior Angles \(= 360^\circ\)
Also:
Exterior Angle \(= 180^\circ −\) Interior Angle
This means the exterior angle and its interior angle form a straight angle.
Example
Find the sum of the exterior angles of a convex pentagon.
A convex pentagon has \(5\) sides and \(5\) exterior angles.
For any convex polygon:
Sum of Exterior Angles \(= 360^\circ\)
So, the sum of the exterior angles of a convex pentagon is:
\(360^\circ\)
This means:
\(\text{Angle } 1 + \text{Angle } 2 + \text{Angle } 3 + \text{Angle } 4 + \text{Angle } 5 = 360^\circ\)
So, the sum of the exterior angles of the convex pentagon is:
\(360^\circ\)
Video Explanation
Practice
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- Sum of Exterior Angles of Regular Polygons (Proof)
- Sum of Exterior Angles of Convex Polygons (Proof) (Current lesson)