ASA/AAS Triangle Solver
Solve a triangle given two angles and one side
Enter Values
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
ASA Triangle
Understanding ASA and AAS Triangles
Both ASA and AAS involve knowing two angles and one side. The difference is the position of the known side relative to the known angles.
ASA vs AAS
ASA (Angle-Side-Angle)
The known side is between the two known angles. Example: Know angles A and B, and side c (which connects them).
AAS (Angle-Angle-Side)
The known side is not between the two known angles. Example: Know angles A and B, and side a (opposite to A).
Why They Give the Same Result
Once you know two angles, you automatically know the third (since A + B + C = 180°). So ASA and AAS both give you all three angles and one side - enough to use Law of Sines to find everything else.
Solving Procedure
- Find the third angle: C = 180° - A - B
- Use Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) = k
- Find unknown sides: a = k·sin(A), b = k·sin(B), c = k·sin(C)
- Calculate area: Area = ½·a·b·sin(C)
Frequently Asked Questions
Is there ever more than one solution?
No! Unlike SSA (the ambiguous case), ASA and AAS always have exactly one solution. Knowing two angles determines the shape, and one side determines the size.
Why is ASA a congruence postulate?
ASA uniquely determines a triangle. If two triangles have ASA equal, they are congruent. This is because the angles fix the shape and the included side fixes the size.
What if angles sum to 180° or more?
If A + B ≥ 180°, no valid triangle exists. The sum of any two angles in a triangle must be less than 180°.
Can I use Law of Cosines instead?
You could, but Law of Sines is simpler for ASA/AAS since you have angles ready to use. Law of Cosines is more natural for SSS and SAS.
What's the difference between ASA and AAS in practice?
Mathematically, they're equivalent - both uniquely determine a triangle. The labels just describe where the known side is relative to the known angles.
Why is AA not enough to determine a triangle?
AA (two angles) determines only the shape (all AA triangles are similar). You need at least one side to determine the actual size.