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3 Responses
Think of the exterior angle at each vertex as the turning you make when you walk around the polygon in one consistent direction (say, counterclockwise, keeping the interior on your left). Along each side you don’t turn at all; all the turning happens at the corners. After you complete one lap and return to your starting point, you must be facing the same way you started, so the total change in your heading is exactly one full turn: 360° (it would be −360° if you walked clockwise). That total turn is the sum of all those exterior “turning angles,” so they add to 360°, whether the polygon is regular, irregular, or concave. At a concave vertex you simply turn the other way, which shows up as a negative exterior angle in the sum, and it still balances out. Simple example: take a concave pentagon with interior angles 60°, 80°, 90°, 100°, and 210° (these add to 540° as they should). The corresponding exterior angles are 120°, 100°, 90°, 80°, and −30°; summing gives 120 + 100 + 90 + 80 − 30 = 360°. The walking/turning viewpoint explains why the total is always one full circle-even for self‑intersecting polygons, the same idea applies.
Imagine walking (or driving) around the polygon: each corner is a little steering-wheel turn by the exterior angle, and by the time you get back you’re facing exactly the way you started, so all those turns add up to one full turn, 360° (or -360° if you go the other way). I might be slightly hand-wavy about sign conventions, but even with concave dents the right-turns (negative) and left-turns (positive) balance so the net turning is exactly one full circle.
Great question! Here’s the clean way to see it: imagine you’re a little “turtle” walking around the polygon in a steady direction (say, counterclockwise). At each vertex you swivel by exactly the amount needed to point along the next edge-that swivel is the exterior angle (take it as positive for left turns, negative for right turns). By the time you’ve traced the whole loop and come back to where you started, you’re also facing the same original direction, which means your total change in heading is one full turn: 360°. That’s true no matter how wiggly, irregular, or concave the polygon is-the individual turns can be big or small, even “right turns” at concave corners, but they always add up to a full spin. For convex polygons there’s also the quick formula proof: sum of exteriors = n·180° − sum of interiors = n·180° − (n−2)·180° = 360°. The key for concave shapes is to think of “exterior angle” as the actual turn you make while walking. Nice visual explanation here: https://www.mathsisfun.com/geometry/exterior-angles-polygons.html. Would you like to test this on a specific concave example (maybe a dart/kite) and add up the signed turns together?