SAS Triangle Calculator
Solve Side-Angle-Side triangles. Enter two sides and the included angle to find all other measurements.
Enter SAS Values
SAS Triangle:
Two sides and the included angle (the angle between the two sides) are known. This always produces a unique triangle.
Solution Method:
1. Use Law of Cosines: c² = a² + b² - 2ab·cos(C)
2. Find angles A and B using Law of Cosines
3. Calculate area and perimeter
Frequently Asked Questions
What is a SAS triangle?
SAS stands for Side-Angle-Side. It's a triangle where you know two sides and the angle between them (the included angle). This is one of the triangle congruence conditions - knowing SAS uniquely determines a triangle.
How do you solve a SAS triangle?
Use the Law of Cosines to find the third side: c² = a² + b² - 2ab·cos(C). Then use the Law of Cosines again (or Law of Sines) to find one of the remaining angles, and finally use the angle sum property to find the last angle.
Why must we use the Law of Cosines for SAS?
The Law of Sines requires knowing at least one angle-side opposite pair. In SAS, we don't have this initially - the known angle is between the two known sides. The Law of Cosines is designed for exactly this situation.
What is an included angle?
An included angle is the angle that lies between two sides. In the SAS case, if you know sides a and b, the included angle is C (the angle at the vertex where sides a and b meet). This is different from a non-included angle.
Can a SAS triangle have no solution?
No, as long as the given values are valid (positive side lengths and an angle between 0° and 180°), a unique triangle will always exist. SAS is a definitive case with no ambiguity, unlike SSA which can have zero, one, or two solutions.
What if the included angle is 90°?
If the included angle is 90°, you have a right triangle and the Law of Cosines simplifies to the Pythagorean theorem: c² = a² + b². You could also use basic trigonometry (sin, cos, tan) to solve it more simply.
How accurate is the SAS calculator?
The calculator uses JavaScript's double-precision floating-point arithmetic, which is accurate to about 15-17 significant digits. Results are displayed to 4 decimal places for sides and 2 decimal places for angles, which is more than sufficient for practical applications.
Can I use SAS for obtuse triangles?
Yes! The SAS method and Law of Cosines work for all types of triangles: acute (all angles less than 90°), right (one angle equals 90°), and obtuse (one angle greater than 90°). The included angle can be any value between 0° and 180°.