Triangle Solver (Two Angles and the Included Side)
The triangulation case the two-sides and three-sides solvers leave.
Example
You enter
- Angle A (deg) 70.89
- Angle B (deg) 49.11
- Included side (baseline) c 9.165
You get
- Third angle C 60 deg
- Side a 10
- Side b 8
Checked against published worked examples from: Naval Education and Training Professional Development and Technology Center (U.S. Navy).
Details, formula, and sources
Scopea measured baseline and the two angles to a point you cannot reach directly. C = 180 - A - B, then the law of sines a = c sin A/sin C and b = c sin B/sin C give the distances to the far point. Sight a target at 70.89 and 49.11 degrees off a 9.165 ft baseline and it is 10.0 and 8.0 ft from the two ends - the same 10-8-9.165 triangle the SAS and SSS solvers build, so all three agree. That is how a surveyor or layout hand reaches a distance to an inaccessible corner, tower, or bank with no tape to the point. The two angles must sum to under 180 degrees. The ambiguous SSA case has its own solver. A layout aid; verify critical dimensions on the work.
Formula or tableC = 180 - A - B; a = c sin(A)/sin(C); b = c sin(B)/sin(C); area = (1/2) a b sin(C). A + B < 180 required.
Source and editionThe law of sines a/sin(A) = c/sin(C) with C = 180 - A - B - first-principles plane trigonometry as in Machinery's Handbook (Industrial Press), by name; public domain.
Free accessPure plane trigonometry, public; the two angles and the baseline are user-supplied measurements.
What governsEstimate. AHJ and licensed professional govern.
API field namesangle_a_deg, angle_b_deg, included_side_c, angle_c_deg, side_a, side_b
- Law of sines C = 180 - A - B, then a = c sin A/sin C, b = c sin B/sin Cplane trigonometry
- Angle sum A + B must be under 180 degrees (the sight lines must meet)geometry