I’m playing with angles in polygons and I love the pattern that exterior angles add up to 360°. It feels so tidy! But I think I’m mixing up definitions when the polygon is concave, and my brain is doing somersaults.
Here’s what I did: I drew a concave hexagon with one “dent” at vertex D. I labeled the exterior angles (the ones that form a straight line with a side, measured on the outside as I walk around clockwise) as 50°, 70°, 60°, 90°, 80°, and x° at the dent. I tried the usual trick: x = 360° − (50 + 70 + 60 + 90 + 80) = 10°. Then I converted back to interior angles: for the five non-dent vertices I used interior = 180° − exterior, so I got 130°, 110°, 120°, 90°, 100°. For the dent, I wasn’t sure – is it interior = 180° − x or 360° − x? If I use 180° − x, that gives 170°, which isn’t reflex. If I use 360° − x, that gives 350°, which explodes the interior angle sum.
Cross-check: a hexagon’s interior angles sum to 720°, and 720° − (130 + 110 + 120 + 90 + 100) = 170°. That matches the 180° − x path, but that means the “dent” wouldn’t be a dent at all. So something’s off in my setup.
My questions: In a concave polygon, does the exterior-angle-sum-of-360° still hold, and if so, which exact definition of “exterior angle” is it using? Do I need to treat the dent’s exterior angle as negative or measured the other way around? How should I properly set up x in this example so the concavity and the sums are consistent?
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3 Responses
Yes-use exterior angles as signed turning angles as you walk around (for clockwise, right turns are negative), so they always add to −360°; here −50 − 70 − 60 − 90 − 80 + x = −360 gives x = −10°, making the interior at D = 170° (so your data actually describe no dent).
If you want D concave, treat that exterior as the opposite sign (a left turn of 10°, equivalently 350° clockwise), giving an interior of 190°.
I love the tidy 360° rule too – the trick is to pin down which “exterior angle” we mean. The version that always works (even for concave polygons) is the turning-angle definition: as you walk around the polygon, the exterior angle at each vertex is the signed turn you make from one side to the next. With a consistent direction, those signed turns add to 360° if you go counterclockwise, and to −360° if you go clockwise. At a concave (reflex) vertex, that signed turn has the opposite sign from the convex ones. If instead you insist on writing all the angles as positive “outside” angles on the walking side, then the total is not 360° anymore – it becomes 360° + 180°×(number of dents).
In your setup you walked clockwise and wrote five positive angles 50°, 70°, 60°, 90°, 80°. Interpreted as signed turning angles, those should actually be −50°, −70°, −60°, −90°, −80° (right turns). To reach the required sum of −360°, the last turn must be −10°, which would make the interior angle at D equal to 180° + (−10°) = 170° – not a dent. In other words, your five numbers already force the last vertex to be convex. If you prefer to keep all six labels positive as “outside-to-the-right” angles, then with one dent the total must be 540°, so x = 540° − (50 + 70 + 60 + 90 + 80) = 190°, and the interior at D is 360° − 190° = 170° again. Same conclusion: those five given angles are incompatible with having a concave vertex; to get a dent, the first five would need to sum to more than 360°.
Quick worked example: take a concave pentagon walked counterclockwise whose interior angles are 90°, 90°, 60°, 90°, 210° (these add to 540°). The signed turning angles are 180 − interior = 90°, 90°, 120°, 90°, −30°. They sum to 360°, and the reflex vertex shows up as the negative turn −30°. If I instead write all the “outside-to-the-left” angles as positive, I get 90°, 90°, 120°, 90°, 150° which sum to 540° = 360° + 180°×1, matching the “one dent” rule. That’s the consistent way to keep both the concavity and the sums playing nicely together.
The 360° fact is true for any simple polygon, convex or concave, provided you use exterior angles as signed turning angles: as you walk around the boundary in order, each exterior angle is the change in your heading, with left turns counted positive and right turns negative (for a clockwise walk, the total is −360°; for counterclockwise, +360°). In terms of interior angle α and signed exterior turn θ, we have α = 180° + θ for a clockwise walk (so convex corners have θ < 0 and reflex ones have θ > 0). Applying that to your data: walking clockwise, the five listed convex corners contribute θ = −50, −70, −60, −90, −80, so the equation is −50 −70 −60 −90 −80 + θ_D = −360, which gives θ_D = −10. That means the “dent” is actually another right turn, and its interior angle is 180 + (−10) = 170°, so the hexagon you set up is in fact convex-exactly what your 720° interior sum check revealed. To truly get a concave hexagon with one reflex vertex, the sum of the five right-turn magnitudes would have to exceed 360°, so that the remaining turn comes out positive; for example, if the five were 50, 70, 60, 90, 100 (sum 370), then θ_D = +10 and the reflex interior would be 190°. If instead you insist on measuring an “exterior angle” as a nonnegative outside wedge at each vertex, then at the dent you must pick the large outside angle (>180°); with that convention the angles no longer add to 360° in general, which is why your numbers “exploded.” I remember tripping over this in a geometry class: I kept adding the cute small outside angles and wondered why my “concave” pentagon magically became convex-switching to signed turns instantly made the 360° rule and the concavity line up.