Formula and Method for Solving a Triangle by Side-Angle-Side (SAS)
When you know two sides of a triangle and the angle trapped between them — the "included angle" — the triangle is fully and uniquely determined. This is the Side-Angle-Side (SAS) case. This calculator takes sides a and b and the included angle C and solves for the missing side c, the two remaining angles A and B, and the triangle's area.
How the calculation works
The Law of Cosines generalizes the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos(C). Taking the square root gives side c, the side opposite the included angle. To find angle A, the calculator applies the Law of Cosines again in its solved-for-angle form: A = arccos((b² + c² − a²) / (2bc)). The third angle follows immediately from the fact that a triangle's interior angles always sum to 180°: B = 180° − A − C. The area comes from a separate identity that needs only the two given sides and the angle between them: Area = ½ · a · b · sin(C). Using the Law of Cosines (rather than the Law of Sines) to find angle A is deliberate — since C is already known to be less than 180°, the remaining two angles cannot include a second obtuse angle, so arccos returns the correct value without the two-solution ambiguity that arcsine can introduce.
Common mistakes
- Using the wrong angle: C must be the angle physically between sides a and b, not an angle at a different vertex — swapping it in changes the whole triangle.
- Angle out of range: the included angle must be strictly between 0° and 180°. A value of 0° or 180° collapses the triangle into a straight line with no area.
- Forgetting degrees vs. radians: this calculator expects the angle in degrees. If your source data is in radians, multiply by 180/π before entering it.
- Mismatched units: enter both sides in the same length unit (both feet, both meters, etc.); the calculator has no way to detect a unit mismatch.
Real-world applications
- Surveying and land measurement use SAS to compute an unknown boundary length from two measured distances and the angle between them.
- Navigation and triangulation find distance and bearing to a target from two known legs and the turn angle.
- Engineering and construction check truss, bracket, or roof-pitch geometry when two member lengths and their angle are fixed by design.
- Physics and vector problems use the same Law of Cosines to find the resultant magnitude when combining two vectors at a known angle.