Why do all exterior angles on a polygon add up to 360 when the interior changes depending on sides?
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Imagine starting at a point in the middle of one of the polygon’s edges and walking one full loop around the polygon’s perimeter. When you return to your starting point, you will have turned a total of 360 degrees and be facing the same direction as when you started.
The exterior angles of a polygon measure the amount of turning that happens at each vertex. For a convex polygon, all turns are in the same direction (either all clockwise or all counterclockwise), so adding up the angles of the individual turns gives you the angle of the total turn you make over the entire loop, which we’ve established is 360 degrees.
For a non-convex polygon, some of the turns are in opposite directions so the exterior angles do not necessarily sum to 360. However, the rule works again if you treat angles in the clockwise direction as negative and angles in the counterclockwise direction as positive, or vice versa. If you track negative angles like this over a path that crosses over itself, you can actually get sums that are larger multiples of 360, like 720 or 1080.
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