SAS Triangle Solver
Solve a triangle given two sides and the included angle
Enter SAS Values
Enter two sides and the angle between them (included angle)
Law of Cosines
c² = a² + b² - 2ab·cos(C)
SAS Triangle
Understanding SAS Triangles
SAS (Side-Angle-Side) means you know two sides and the angle between them (the included angle). This configuration uniquely determines a triangle - there's always exactly one solution.
Solving SAS Triangles
- Find the third side: Use Law of Cosines: c² = a² + b² - 2ab·cos(C)
- Find another angle: Use Law of Cosines or Law of Sines
- Find the last angle: Use A + B + C = 180°
- Calculate area: Area = ½ × a × b × sin(C)
Why SAS is Unambiguous
Unlike SSA (the ambiguous case), SAS always has exactly one solution. The included angle "locks" the triangle into a unique shape. You can visualize this: if you fix two sides at a specific angle, there's only one way to connect their endpoints.
Area Formula for SAS
When you have two sides and the included angle, area is straightforward:
Frequently Asked Questions
Why must the angle be between the two sides?
The "included" angle is crucial. If the angle isn't between the two known sides, you have SSA (ambiguous case) instead, which can have 0, 1, or 2 solutions.
Can I use Law of Sines for SAS?
Not to start. Law of Sines requires a side-angle pair. Use Law of Cosines first to find the third side, then you can use Law of Sines for the remaining angles.
What if the included angle is 90°?
Then cos(C) = 0, and Law of Cosines simplifies to the Pythagorean theorem: c² = a² + b². It's a right triangle!
Is there always a valid triangle?
Yes, as long as the included angle is between 0° and 180° (exclusive). Any positive side lengths with a valid angle will form a triangle.
How is SAS used to prove congruence?
SAS is a triangle congruence postulate: if two triangles have two pairs of equal sides with equal included angles, the triangles are congruent.
Why use Law of Cosines instead of Law of Sines?
Law of Cosines directly handles SAS. Law of Sines would introduce the ambiguous case when finding angles from sides.