Exterior Angle Sum Theorem for Convex Polygons

In any convex polygon with n sides, the sum of the exterior angles always adds up to a full circle (360°).

In other words, the total of the exterior angles in a convex polygon remains the same, no matter how many sides it has.

For instance, whether it’s a triangle, a square, a pentagon, or any polygon with n sides, the sum of the exterior angles will always be 360°.

Example: Imagine you’re walking along the edge of a fence. The exact shape of the fence doesn’t matter—it could have many angles or just a few. The only condition is that it must be a convex polygon. Every time you reach a corner, you change direction.

The total of all these direction changes as you walk around the fence is exactly 360°, just as if you had made a complete circle around it.

The Proof

Let’s consider a convex polygon.

For example, a convex polygon with n = 5 sides.

Each exterior angle of a convex polygon is adjacent to an interior angle.

Therefore, each exterior angle is supplementary to its adjacent interior angle, together forming a straight angle of 180°.

This means that in a convex polygon with n sides, there are n pairs of angles, and the sum of the interior and exterior angles together is n × 180°.

$$ n \cdot 180° $$

Knowing that the sum of the interior angles is equal to (n-2) × 180°, we can calculate the sum of the exterior angles by subtracting this from the total.

In essence, the sum of the exterior angles is equal to the total of both the interior and exterior angles minus the sum of the interior angles.

Thus, the sum of the exterior angles of a convex polygon is:

$$ n \cdot 180° - [ (n-2) \cdot 180° ] $$

With a bit of simple algebra, this simplifies to:

$$ [ n - (n-2) ] \cdot 180° = $$

$$ ( n - n + 2 ) \cdot 180° = $$

$$ 2 \cdot 180° = 360° $$

So, no matter how many sides a convex polygon has, the sum of its exterior angles will always be 360°.

And that’s the theorem!