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Sum of Exterior Angles
At each vertex, extend one side past the vertex and measure the angle to the next side — that's the exterior angle there. Add up all of them, for any simple polygon, and you always get:
Sum of exterior angles = 360°
That total never changes, no matter how many sides the polygon has.
Why it's always 360°, not (n−2)×180°
Walk all the way around the polygon's boundary back to your starting point, facing your starting direction. At each vertex you turn through that exterior angle. By the time you're back where you started, you've turned through exactly one full rotation — 360° — regardless of n.
Quick checks
Square: four 90° turns = 360°. Regular hexagon: six 60° turns = 360°. Regular decagon: ten 36° turns = 360°.
Concave vertices turn the other way
At a "caved-in" (reflex) vertex, you turn the opposite direction as you walk the boundary, so that exterior angle counts as negative. The total still comes out to exactly 360° once the negative ones are added in. Drag a vertex inward until the badge reads "Concave" to see one appear.
How this differs from the interior angle sum
The interior sum, (n − 2) × 180°, grows as you add sides. The exterior sum never does — it's a constant 360°. The two facts are two sides of the same coin: interior + exterior = 180° at every vertex.
One thing to watch for
This assumes a simple polygon — sides that don't cross. Cross one and the measured total drifts from 360°, which the tool will flag.
A bit of history
It's the same triangulation-style reasoning behind Euclid's Elements (Book I), restated: walking any closed simple loop once around always turns you through exactly one full revolution.