Ambiguous Case Calculator (SSA)
Analyze the SSA case to determine 0, 1, or 2 triangle solutions
Enter SSA Values
Given: Two sides and an angle opposite one of them
SSA Case Analysis
Understanding the Ambiguous Case (SSA)
The SSA (Side-Side-Angle) case is called "ambiguous" because knowing two sides and an angle opposite one of them doesn't uniquely determine a triangle. There could be 0, 1, or 2 valid triangles.
When Angle A is Acute (< 90°)
Calculate height: h = b × sin(A)
- a < h: No solution (side too short)
- a = h: One solution (right triangle)
- h < a < b: Two solutions (ambiguous!)
- a ≥ b: One solution
When Angle A is Obtuse (≥ 90°)
- a ≤ b: No solution
- a > b: One solution
Why Two Solutions?
When using Law of Sines to find angle B, we get sin(B) = (b × sin(A))/a. Since sin(B) = sin(180° - B), there are potentially two angles that work: B and (180° - B). Both create valid triangles if C > 0.
Frequently Asked Questions
Why is SSA called the ambiguous case?
Because knowing two sides and an angle opposite one of them doesn't uniquely determine a triangle - there might be 0, 1, or 2 possible triangles.
How do I know if there are two solutions?
When angle A is acute and h < a < b (where h = b sin A), there are two solutions. Both B and (180° - B) give valid angles.
What is the height h used for?
The height h = b sin(A) is the perpendicular distance from vertex B to the base. If side a is shorter than h, it can't reach the base to form a triangle.
Why is there no ambiguity with SAS or ASA?
SAS and ASA uniquely determine triangles because the known elements are in positions that lock down the shape. SSA doesn't constrain the triangle the same way.
Can I use Law of Cosines instead?
Law of Cosines requires SAS or SSS. For SSA, you must use Law of Sines, which introduces the ambiguity through the inverse sine function.
How do I find both solutions when there are two?
Find B₁ = arcsin((b sin A)/a). Then B₂ = 180° - B₁. Check if C₁ = 180° - A - B₁ and C₂ = 180° - A - B₂ are both positive.
What happens when a = h exactly?
When a = h = b sin(A), there's exactly one solution, and it's a right triangle with angle B = 90°.
Is SSA ever unambiguous?
Yes! When a ≥ b (the side opposite the given angle is at least as long as the adjacent side) or when angle A is obtuse and a > b.