How this instrument works
An exterior angle appears when one side of a triangle is stretched past a vertex, and it sits supplementary to the interior corner right beside it — the two share a straight line, so they always total 180°. What makes this figure worth its own instrument is a second, less obvious fact: it also equals the sum of the two interior corners at the OTHER vertices, the ones geometers call remote because they don't touch the corner in question. Extend the side past C and what opens up there equals A + B, full stop, no C required.
The derivation takes one line once you already know a triangle's three corners sum to 180°: the exterior measure at C is 180° − C, and C itself is 180° − A − B, so substituting gives 180° − (180° − A − B), which collapses to A + B exactly. Euclid recorded this as Proposition 32 of the Elements, but a weaker cousin appears sixteen propositions earlier: Proposition 16 shows that same exterior measure merely exceeds each remote corner individually, a fact provable without assuming parallel lines behave the way Euclid's fifth postulate says they do. The equality in Proposition 32 needs that postulate; the inequality in Proposition 16 does not.
One boundary is built into the geometry itself: A and B must sum to less than 180°, because the third corner C = 180° − A − B has to stay positive for the triangle to exist at all. Push A + B toward that ceiling and the exterior measure balloons toward a full straight line, 180°, while C is squeezed toward zero — the triangle flattens at exactly the moment the formula stops making sense.
- Enter the first known corner into Interior angle A — degrees by default, with radians and turns also selectable.
- Enter the second known corner into Interior angle B, using whichever matching unit you measured it in.
- Read Exterior angle at C for the measure formed by extending the side opposite that pair, past the third vertex.
- Keep A + B under 180°; anything at or past that limit leaves no positive measure for C, so the triangle can't close.
Worked example — a 50°, 60° triangle
A triangle has interior corners of 50° and 60° at two of its vertices, and a side is extended past the third one. Interior angle A takes 0.8726646259971648 radians (50°) and Interior angle B takes 1.0471975511965976 radians (60°); the sheet returns Exterior angle at C = 1.9198621771937625 radians, which is exactly 110°.
Checking it the long way confirms the shortcut: the third corner is 180° − 50° − 60° = 70°, so the exterior measure beside it is 180° − 70° = 110° — the same result the direct addition gave. Either route lands on 110° because the two paths are the same statement in different clothes; the calculator simply skips the middle step and adds 50° + 60° straight through.
Questions
What is the exterior angle theorem for a triangle?
It states that an exterior angle of a triangle equals the sum of the two remote interior corners — the ones it doesn't touch. Extend a side past vertex C and that exterior measure equals A + B, the pair at the other two corners, regardless of what C itself measures.
Why does the exterior angle equal the sum of the two remote angles?
Because two separate facts combine into one. The exterior measure and its adjacent corner C sit on a straight line, so they sum to 180°; and the triangle's own corners sum to 180° too, giving C = 180° − A − B. Substituting the second fact into the first cancels the 180° and C terms, leaving exterior = A + B exactly.
Is the exterior angle theorem the same as Euclid's Proposition 32?
Yes, that equality is Proposition 32 of Euclid's Elements, and it relies on the parallel postulate. A weaker, older result — Proposition 16 — shows only that the exterior measure exceeds each remote corner, which is true even in geometries where the parallel postulate fails.
What mistake do people usually make with this formula?
Adding the wrong pair — including the near vertex's own interior corner instead of the two remote ones. The exterior measure at C ignores C entirely and depends only on A and B; swapping in C for one of them gives a number that has no geometric meaning for that vertex.
Do all three exterior angles of a triangle share a pattern?
Yes — taken one per vertex, they always sum to exactly 360°, no matter the triangle's shape. That's not a coincidence unique to triangles either: the exterior measures of any convex polygon, walked around once, always total 360°, the same full turn a walker makes circling the shape.
Can an exterior angle be a right angle or larger?
Yes. A right exterior measure of exactly 90° needs A + B = 90°, which happens whenever the triangle's third corner C is itself 90°. Push A + B further and the exterior measure grows past 90° and can approach, but never reach, the full 180° that would flatten the triangle.