Triangle Solver (Two Sides and a Non-Included Angle, the Ambiguous Case)
Two sides and an angle that is NOT between them can fit zero, one, or two triangles, and this says which.
Example
You enter
- Angle A (deg) 49.11
- Side a (opposite angle A) 8
- Side b (from angle A's vertex) 10
You get
- Solutions 2
- Triangle 1 (angles B and C, third side) 70.9
- Side c 9.165
- Angle b2 (deg) 109.1
- Side c2 3.929
No published worked example is on file for this calculation. It is checked against a hand-worked derivation of its method and an independent reference test.
Details, formula, and sources
ScopeThe law of sines gives sin B = b sin A / a; with A acute, compare side a against the height b sin A. Shorter and it cannot reach; equal and it is one right triangle; between the height and b and it swings to two places, a long and a short third side; b or longer and there is one. A 49.11 degree angle with 8 opposite and 10 beside it fits the 10-8-9.165 triangle the other solvers build and also a 3.93 short one, so one more measurement settles which you have. With A at 90 degrees or more, one triangle only when a is longer than b. A layout aid; verify critical dimensions on the work.
Formula or tableh = b sin(A). A < 90: a < h none; a = h one (B = 90); h < a < b two (B1 = asin(b sin A / a), B2 = 180 - B1); a >= b one. A >= 90: one if a > b, else none. C = 180 - A - B; c = a sin(C)/sin(A); area = (1/2) a b sin(C).
Source and editionThe law of sines a/sin(A) = b/sin(B) with the ambiguous-case (SSA) test against the height b sin A - first-principles plane trigonometry as in Machinery's Handbook (Industrial Press), by name; public domain.
Free accessPure plane trigonometry, public; the two sides and the angle are user-supplied measurements.
What governsEstimate. AHJ and licensed professional govern.
API field namesangle_a_deg, side_a, side_b, solutions, angle_b_deg, side_c, angle_b2_deg, side_c2
- Law of sines sin B = b sin A / a; C = 180 - A - B; c = a sin C / sin Aplane trigonometry
- Ambiguous-case test count of triangles from a against the height b sin A and against b (A acute), or a against b (A right or obtuse)plane trigonometry