Formula and method for the AAA Triangle Calculator
An AAA (angle-angle-angle) problem gives you all three interior angles of a triangle. On their own, three angles only fix the triangle's shape, not its size — infinitely many similar triangles share the same three angles at different scales, which is why AAA proves similarity rather than congruence. This calculator applies the triangle angle-sum rule and the Law of Sines to size an actual triangle once you also supply one known side length.
How the calculation works
First, the calculator checks that the three angles you entered add up to exactly 180°, the fixed interior angle sum of every triangle on a flat plane. If they do, and you've supplied side a (the side opposite Angle A), it applies the Law of Sines — a / sin(A) = b / sin(B) = c / sin(C) — to solve for sides b and c. The triangle's area then follows from Area = ½ × a × b × sin(C), using the two known sides and the angle between them.
Common mistakes
- Angles that don't sum to 180°: if your three angles add up to anything else, no such planar triangle exists — double-check your measurements.
- Confusing AAA with SSS or SAS: AAA proves similarity (same shape), not congruence (same shape and size). Two AAA triangles can look identical but be different sizes; you need at least one side to pin down actual lengths.
- Mismatched side/angle pairing: side a must be the side opposite Angle A, not adjacent to it — swapping this breaks the Law of Sines ratio and gives wrong side lengths.
Real-world applications
- Surveying and triangulation, where distant angles are measured but only one baseline distance is known
- Map scaling and navigation, converting bearing angles into distances once one reference length is fixed
- Engineering and architecture, checking that a designed triangle's angles are consistent before scaling it to real dimensions
- Trigonometry education, illustrating why AAA determines similarity but not congruence