What Is the Exterior-Angle Sum?

Why the total is 360 degrees

A useful way to understand the rule is to imagine walking around the polygon. Each exterior angle is a turning amount, and one complete trip around the boundary brings you back to your starting direction after a full turn.

This fact works for regular and irregular polygons. In a regular n-gon, it becomes especially simple because each exterior angle has the same measure, so each one is three hundred sixty divided by n.

At each corner, the direction changes by an outside turn before the walk continues along the next side. After the final side, you are facing the original direction again, so the total change must be one complete turn. This is why the sum stays fixed even as the number of corners changes.

• The exterior angles of a polygon always add to 360 degrees.
• The sum of one exterior angle per vertex is always three hundred sixty degrees.
• The rule works for regular and irregular polygons when the exterior angles are taken consistently around the shape.
• In a regular polygon, each exterior angle equals 360 divided by n.

When the outside angles help

• Use the exterior-angle sum rule to find one exterior angle of a regular polygon.
• Use it to solve for the number of sides in regular polygon problems.
• Use it to connect polygon geometry with the idea of a full turn or rotation.

Exterior-sum errors to catch

• Do not add both possible exterior angles at each vertex; use one consistent set around the polygon.
• Do not confuse the exterior-angle sum with the interior-angle sum, which depends on n.
• Do not assume the rule gives one exterior angle unless the polygon is regular.