Triangle Solver (Two Sides and the Included Angle)

Two sides and the angle between them: the third side by the law of cosines c = sqrt(a^2 + b^2 - 2ab cos C), plus the other two angles and the area (1/2) ab sin C. Sides 10 and 8 at 60 degrees give a third side of 9.165 and corners of 70.9 and 49.1 degrees; set the angle to 90 and the same sides give exactly 12.806 - the Pythagorean hypotenuse - so the general solve collapses to a square corner. For a corner brace, a hip or valley rafter, a guy from a known spread, or a survey leg. SSS, ASA/AAS, and the ambiguous SSA are separate. A layout aid; verify critical dimensions on the work.

c = sqrt(a^2 + b^2 - 2ab cos C); A = acos((b^2 + c^2 - a^2)/(2bc)); B = 180 - C - A; area = (1/2) ab sin C. C = 90 gives c = sqrt(a^2 + b^2).

The law of cosines c^2 = a^2 + b^2 - 2ab cos(C) with the SAS area (1/2) ab sin(C) - first-principles plane trigonometry as in Machinery's Handbook (Industrial Press), by name; public domain.

Pure plane trigonometry, public; the two sides and the included angle are user-supplied measurements.

Estimate. AHJ and licensed professional govern.

Field names used by the API: side_a, side_b, included_angle_deg, side_c, angle_a_deg, area

• Law of cosines c = sqrt(a^2 + b^2 - 2ab cos C); collapses to Pythagoras at C = 90plane trigonometry
• SAS only two sides + included angle; SSS, ASA/AAS, and ambiguous SSA are separatescope of this tile