I keep tripping over angles in polygons, especially when the shape isn’t a nice, regular hexagon from a poster. I know the textbook rules: sum of interior angles is (n−2)*180, exterior angles add to 360 if you walk around the shape, yada yada. Sounds easy until the diagram has one of those “pushed-in” corners and a random x drawn outside the shape.
Personal horror story: last year I tanked a quiz because I kept mixing up interior vs exterior and when to use 360. Thought I had it cleaned up since, but I just hit a heptagon problem and my brain short-circuited again. The picture shows a 7‑sided polygon, not regular. They give a bunch of interior angles: 110°, 120°, 95°, and two right angles (90°, 90°). One corner looks concave (like it’s dented inward), and the angle they want is x, which they drew outside at that dented corner. I figured: total interiors should be (7−2)*180 = 900. So I did 900 − (110+120+95+90+90) to get what’s left for the last two corners. Then I tried telling myself, “Okay, at the dented corner the interior is 360 − something, so the exterior is… uh, 180 − something? Or is x actually the reflex interior, not the exterior?” I ended up calling x supplementary to the interior there and got a number that the answer key hates.
What I need is a dead-simple way to keep this straight in my head: when do I use 900 vs 360, and how do I tell at a glance if the labeled outside angle is the exterior angle I’m supposed to use or just the outside part of a straight line at that vertex? Also, for concave polygons, does the (n−2)*180 rule still hold exactly the same, and if so, what’s the quick relationship for x at that inward corner without redrawing everything? Bonus points for a mental trick that doesn’t involve 12 cases and three definitions.
Any help appreciated!
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3 Responses
Short version to keep it straight: use 900° (that is, (n−2)·180° with n=7) whenever you’re adding interior angles to find a missing interior; use 360° only when you are summing all the exterior “turning” angles as you walk once around the polygon. The interior-sum formula works for concave polygons too. At a single vertex, decide what “outside angle” x is by how it sits: (1) If x is drawn on the straight extension of a side (a linear-pair picture), then x = |180° − I|, where I is the interior angle at that vertex. So for a convex corner x = 180° − I, and for a concave “pushed‑in” corner x = I − 180°. (2) If x is the outside wedge between the two sides (so x and I together make the full 360° around the point), then x = 360° − I. A quick test: does x with I look like a straight line? Use 180. Do they fill the whole circle at the vertex? Use 360. For your heptagon, the five given interiors sum to 505°, so the last two interiors total 395°; once you identify the dented interior I, take x = I − 180° (linear style) or x = 360° − I (outside‑wedge style). Alternatively, convert each interior to a turning exterior E = 180° − I (it’s negative at the concave corner), sum all E’s to 360°, and then x is |E| at that vertex. Without your exact diagram I can’t pin a number, but these rules cover the standard cases. A visual recap with examples is here: https://www.khanacademy.org/math/geometry/hs-geo-foundations/hs-geo-angles/a/exterior-angles-of-polygons
Pocket rule: (n−2)·180 works for any simple polygon (even with a “dent”); use it to find the interior at that vertex, then decide x locally-if x is the little outside wedge between the two sides at a concave corner, x = 360 − interior there, but if x is the angle outside forming a straight line with a side, x = 180 − that interior (the 360 sum is only for the full walk-around of all exterior turns).
Example: concave pentagon with four interiors 90°, 80°, 70°, 60° ⇒ total 540°, so dent interior = 540 − 300 = 240°, hence the outside wedge x = 360 − 240 = 120°.
A tidy way to keep it straight: for any simple n-gon (convex or concave) first use (n−2)*180 to find the interior(s) (900 for n=7), then decide x by “180 with a line, 360 around a point”-if x sits on a straight extension (a linear pair) then x = 180 − interior, but if x is the small outside wedge at a pushed‑in (concave) corner between the two sides then x = 360 − interior; the “sum to 360” rule applies only to the one turning exterior angle at each vertex as you walk around.
Example: if the dented interior works out to 230° from the 900° total, then x = 360 − 230 = 130° (whereas at a convex corner with interior 110°, a linear‑pair outside would be 180 − 110 = 70°); see https://www.khanacademy.org/math/geometry/hs-geo-foundations/hs-geo-angles/a/interior-and-exterior-angles-of-polygons.