The sum of exterior angles of any polygon is equal to

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The sum of exterior angles of any polygon is equal to

As you walk around a polygon and measure the angle you turn at each vertex, those turning angles add up to a full turn. Moving from one side to the next, you keep turning in the same direction until you complete the loop and end up facing the direction you started with. That complete turn is 360 degrees, so the sum of the exterior angles around any polygon is 360 degrees. You can also see this by relating exterior angles to interior angles: each exterior angle is the angle you would turn if you extend the sides and look at the outside angle, which is 180 degrees minus the corresponding interior angle in a convex case. Adding these around the polygon gives n·180 minus the sum of interior angles, which is (n·180) − [(n−2)·180] = 360 degrees. This reasoning holds for simple polygons when you think of exterior angles as the turning angles as you traverse the boundary. So the total exterior-angle sum is 360 degrees.

As you walk around a polygon and measure the angle you turn at each vertex, those turning angles add up to a full turn. Moving from one side to the next, you keep turning in the same direction until you complete the loop and end up facing the direction you started with. That complete turn is 360 degrees, so the sum of the exterior angles around any polygon is 360 degrees.

You can also see this by relating exterior angles to interior angles: each exterior angle is the angle you would turn if you extend the sides and look at the outside angle, which is 180 degrees minus the corresponding interior angle in a convex case. Adding these around the polygon gives n·180 minus the sum of interior angles, which is (n·180) − [(n−2)·180] = 360 degrees. This reasoning holds for simple polygons when you think of exterior angles as the turning angles as you traverse the boundary.

So the total exterior-angle sum is 360 degrees.

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