I’m trying to tame the wild herd of angles inside and outside polygons, but they keep galloping in circles. I think I know two things: (1) the sum of interior angles of an n-gon is (n−2)×180°, and (2) in a regular n-gon each exterior angle is 360°/n. My head nods yes… until I draw a not-so-regular shape and it screams no.
Here’s where I wobble. If I have a non-regular pentagon and I label the exterior angles around it as 40°, 70°, 100°, 80°, and x, I tried the equation 40 + 70 + 100 + 80 + x = 360°. That gives me a value for x, but my sketch then gives interior angles that don’t add up to (n−2)×180°. I suspect I’m measuring some of those “exteriors” the wrong way (maybe going the reflex way around a corner?).
Also, for a hexagon with one “caved-in” vertex (a concave hexagon), say the interior angle at that dent is 230°. If I walk around the outside and mark exterior angles, does the 360° total still apply? Do I have to think of that dented corner’s exterior as negative or measured the other direction?
Finally, tiny sanity check: for a regular nonagon I get each interior angle as (9−2)×180°/9 = 140°. I also notice 180° − 360°/9 = 140°. Are those two ways always equivalent, or am I accidentally doing a lucky algebra dance?
Could someone show me the right way to define and add exterior angles so I stop mixing them up with interior ones, especially for shapes with a dent? And in my pentagon example, how should I set up x correctly without breaking the interior-angle sum?
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3 Responses
Ohhh, I’ve done exactly this dance with exterior angles – and yes, I’ve tripped over the same corner where a “nice” outside angle quietly turns into a reflex or slips inside without telling me. So you’re in very good company.
Here’s how I keep it straight in my head (on a good day): think of walking a little toy car around the edges of the polygon. At each vertex, the “exterior angle” is how much you turn your steering wheel to face the next side. If all your turns are gentle left turns as you cruise around the shape once, you’ve used up a total of 360° of left turn by the time you get back where you started. If you ever have to make a right turn (that’s what a “dent” forces you to do), that right turn cancels part of your left-turn budget.
That leads to three practical rules I use.
1) Convex polygons (no dents)
– Use the steering-wheel exterior angles – the ones that actually turn you from one side to the next, not the big reflex-y arcs or the interior angles by accident.
– When you do that consistently, the exterior angles add to exactly 360°.
– This does not require regularity; irregular but convex is totally fine. The exterior angles can all be different – they still add to 360° as long as you’re measuring the “turns.”
2) Concave polygons (with dents)
– A dent (interior angle bigger than 180°) forces a right turn at that vertex.
– If the interior angle at the dent is, say, 230°, then the amount you “turn the wrong way” is 230° − 180° = 50°. That 50° comes off your 360° steering budget.
– So for one dent of 230°, the sum of the (steering) exterior angles is 360° − 50° = 310°.
– If there are multiple dents, you subtract the “excess over 180°” for each one.
3) Don’t mix definitions mid-walk
– The most common way to get nonsense totals is to mix: (a) a genuine steering turn at one corner, (b) a reflex outside arc at another, and (c) an interior angle masquerading as an exterior at a third. Then 360° goes out the window because you’re not summing the same kind of thing each time.
A tiny analogy: imagine keeping a spin ledger as you walk around. Every left turn you write down as positive; every right turn as negative. If there are no dents, your positives add to +360°. If you have a 50° right turn somewhere, your ledger ends at 360° − 50° = 310°. Same idea.
Now, to your specific bits:
Your irregular pentagon with exterior angles 40°, 70°, 100°, 80°, and x
– One of those (the 100°) is suspicious in the sense that it looks like you may have written the interior at that corner instead of the steering exterior – this is super easy to do when sketching.
– The steering exterior is the supplement of the interior only when you’re turning the correct way. If what you wrote (100°) is actually the interior, then the exterior turn you meant is 180° − 100° = 80°.
– So I would set it up as 40 + 70 + 80 + 80 + x = 360, which gives x = 90°.
– Check against interiors: the corresponding interior angles would be 140°, 110°, 100°, 100°, 90°, which do add to (5 − 2) × 180° = 540°. So the interior-angle sum behaves itself again.
Your concave hexagon with a dent of 230°
– The “excess over straight” is 230° − 180° = 50°. That’s a right turn in the steering sense.
– So the total of the (steering) exterior angles around the hexagon is 360° − 50° = 310°.
– If you prefer to keep everything positive by measuring the big outside swing at the dent, that’s fine, but then you’re not summing steering turns anymore, so don’t expect 360°.
Sanity check on the regular nonagon
– For a regular n‑gon, both views line up: each interior angle is (n − 2) × 180° / n, and also 180° − 360° / n. For a regular nonagon that’s 140° either way.
– That equivalence relies on the exterior angles all being equal (360°/n) – that is, on regularity/equiangularity. For a not-regular polygon, 180° − 360°/n no longer describes any one interior angle; at best it’s the “average,” and you shouldn’t expect individual angles to match it.
How to keep exterior angles straight, especially with dents
– Decide you’re measuring steering turns. Walk around in one direction (say, counterclockwise), marking at each corner the angle you actually turn to align with the next side.
– For convex vertices, that’s a left (positive) turn; for a dent, it’s a right (negative) turn whose size is interior − 180°.
– Add them with their signs. If there are no dents, you’ll get 360°. With dents, subtract the right turns from 360°.
– If any number you wrote was actually an interior angle by mistake, replace it with the corresponding steering turn before summing. That’s what fixed the pentagon.
I know this seems fiddly, but once you stick to “how much did I turn my steering wheel at each corner?” the arithmetic behaves and the sketches stop arguing back. And if your sketch still insists on disagreeing, it’s usually because one of the angles was recorded the reflex way or as an interior in disguise – I still catch myself doing that and then wondering why my total went wonky!
I find it easiest to fix a walking rule first: trace the polygon counterclockwise, and at each vertex record the signed turn needed to face along the next side. Take left turns as positive, right turns as negative, and always use the smaller turn in magnitude (between −180° and 180°). With that definition, the exterior angles of any simple polygon-convex or concave-sum to 360° (it would be −360° if you walked clockwise), because your facing direction completes exactly one full rotation. The relation interior + exterior = 180° still holds with signed exteriors: at a “dent” with interior 230°, the exterior is 180° − 230° = −50°. So yes, for your concave hexagon the 360° total still works, provided you keep −50° at the dent. If you instead record a reflex outside angle like 310° at that corner, convert it to its signed turn −50° first; otherwise the totals won’t match and the interior-angle sum will look wrong. A clear visual rundown is here: https://www.mathsisfun.com/geometry/exterior-angles-polygons.html
Now to your pentagon: if the 40°, 70°, 100°, and 80° are signed exterior turns from a single counterclockwise walk, then x = 360° − (40 + 70 + 100 + 80) = 70°. The interiors then are 180° − each exterior: 140°, 110°, 80°, 100°, 110°, which add to 540° as expected. If your sketch gives a different interior total, it likely means one “exterior” was taken the long way around (say 260°) instead of the signed −100°; replace it by the signed value before summing. And your nonagon sanity check is always valid: (n − 2) × 180°/n = 180° − 360°/n, so those two interior-angle formulas agree for every regular n-gon.
The key is to use one consistent definition of “exterior angle”: walk once around the polygon with the interior on your left (counterclockwise), and at each vertex take the signed turning angle from one side to the next, measured in the range (−180°, 180°]. With this convention the exterior angles of any simple polygon (convex or concave) always sum to 360°; if you walk the other way they sum to −360°. At a convex vertex with interior angle θ the turning angle is 180° − θ (a left turn, positive). At a concave vertex (θ > 180°) it is 180° − θ, which is negative (a right turn). If instead you sometimes pick the reflex outside angle, the totals won’t come out to 360° because you’ve changed conventions midstream. Worked example: your pentagon with exterior angles 40°, 70°, 100°, 80°, x gives x = 360 − (40 + 70 + 100 + 80) = 70°. The interior angles are then 180 − each exterior: 140°, 110°, 80°, 100°, 110°, which add to 540° = (5 − 2) × 180°. For a concave hexagon with a “dent” of 230° inside, the exterior at that vertex is 180 − 230 = −50°; the six exteriors still total 360°, so the other five must add to 410°. And your nonagon check is right: (n − 2) × 180°/n = 180° − 360°/n by simple algebra. A clear summary of the 360° rule is here: https://en.wikipedia.org/wiki/Exterior_angle.