Formula and Method: The Triangle Sum Theorem
The Triangle Sum Theorem states that the three interior angles of any triangle — no matter its shape or size — always add up to 180° (π radians). If you know any two angles, you can find the third with simple subtraction: C = 180° − A − B. This calculator also classifies the triangle by its angles (acute, right, or obtuse) and by its angle symmetry (equilateral, isosceles, or scalene).
How the calculation works
Enter two of the triangle's three interior angles and choose whether they're measured in degrees or radians. The calculator subtracts both known angles from a straight angle — 180° in degree mode, or π (≈3.1416 rad) in radian mode — to find the third angle: C = 180° − A − B. The theorem holds for every triangle in Euclidean (flat) plane geometry, whether it's a tiny sliver or an enormous plot, right or oblique.
Common mistakes
- Angles that don't leave room for a third: if A + B is already 180° or more, there's zero or a negative measure left for C, which isn't a valid triangle.
- Mixing degrees and radians: 180° equals π radians, so switch the unit selector rather than converting by hand — entering a radian value while in degree mode (or vice versa) gives a nonsensical result.
- Confusing the Triangle Sum Theorem with the Exterior Angle Theorem: the Triangle Sum Theorem says interior angles total 180°; the Exterior Angle Theorem says an exterior angle equals the sum of the two remote interior angles. They're related but answer different questions.
Real-world applications
- Surveying and navigation use known bearing angles and the theorem to solve for an unmeasured angle in a triangulated plot.
- Carpentry and framing rely on the 180° rule to cut miters and rafters that close a triangular frame cleanly.
- Trigonometry and physics problems often need the third angle before applying the Law of Sines or Law of Cosines.
- Geometry proofs use the theorem as a foundation for the exterior angle theorem, polygon angle-sum formulas, and triangle similarity arguments.