Explanation
In this lesson, we prove that the sum of the exterior angles of a regular polygon is always:
\(360^\circ\)
An exterior angle is formed outside a 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
For regular polygons, all sides are equal and all interior angles are equal. This means all exterior angles are equal too.
When we go around a regular polygon, the exterior angles represent a full turn.
A full turn is:
\(360^\circ\)
So, the sum of the exterior angles of a regular polygon is always:
\(360^\circ\)
This works for a regular triangle, square, regular pentagon, and any regular polygon.
Formula / Rule
For any regular polygon:
Sum of Exterior Angles \(= 360^\circ\)
For each exterior angle of a regular polygon:
Each Exterior Angle \(= \frac{360^\circ}{n}\)
where:
\(n\) is the number of sides of the regular polygon.
Also:
Exterior Angle \(= 180^\circ −\) Interior Angle
Example
Find the sum of the exterior angles of a square.
A square is a regular quadrilateral.
It has \(4\) equal exterior angles.
Since the sum of the exterior angles of a regular polygon is always:
\(360^\circ\)
we have:
Sum of Exterior Angles \(= 360^\circ\)
To find each exterior angle:
Each Exterior Angle \(= \frac{360^\circ}{4}\)
\(= 90^\circ\)
So, each exterior angle of a square is:
\(90^\circ\)
And the sum of all exterior angles is:
\(90^\circ + 90^\circ + 90^\circ + 90^\circ = 360^\circ\)
Video Explanation
Practice
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- Sum of Exterior Angles of Regular Polygons (Proof) (Current lesson)
- Sum of Exterior Angles of Convex Polygons (Proof)