Formula and Method for the SAS Triangle Calculator
SAS stands for Side-Angle-Side: you know two sides of a triangle, a and b, and the measure of the angle C sandwiched between them. That is enough information to pin down the triangle completely — its third side, its other two angles, its area, and its perimeter all follow from a and b and C using the Law of Cosines and a companion area formula. This calculator applies those formulas directly, so you don't have to solve them by hand.
How the calculation works
The third side, c, comes from the Law of Cosines: c² = a² + b² − 2ab·cos(C). This is a generalization of the Pythagorean theorem that works for any triangle, not just right triangles — when C = 90°, cos(C) = 0 and the formula collapses to c² = a² + b², the familiar Pythagorean relationship. Once c is known, angle A (opposite side a) is found by rearranging the Law of Cosines: cos(A) = (b² + c² − a²) / (2bc), then take the inverse cosine. The last angle follows immediately from the triangle angle sum: B = 180° − A − C. The area comes from a separate formula that only needs the two given sides and the included angle: Area = ½ · a · b · sin(C), and the perimeter is simply a + b + c.
Common mistakes
- Using the wrong angle: C must be the angle physically between sides a and b (the included angle), not an angle adjacent to only one of them. If you plug in the wrong angle, every downstream result is wrong.
- Degrees vs. radians: this calculator expects the angle in degrees. If your source data is in radians, convert first (multiply by 180/π) or the sine and cosine values will be off.
- Angle out of range: C must be strictly between 0° and 180° — a triangle cannot have an interior angle at or beyond a straight line.
- Mixing units: enter both side a and side b in the same unit; the third side, area, and perimeter are reported in that same unit (or its square, for area).
Real-world applications
- Surveying and land measurement use SAS when two boundary lengths and the angle between them are known but a direct distance is not.
- Navigation and dead-reckoning use SAS to find resultant distance and bearing from two legs of a course and the turn angle between them.
- Engineering and construction use SAS to check truss, brace, or roof-pitch geometry when two member lengths and their joint angle are specified.
- Trigonometry and geometry coursework use SAS as the standard entry point for the Law of Cosines and triangle-solving techniques.