What this calculator does
The ambiguous case comes up when a triangle is defined by two sides and a non-included angle (SSA): one angle, the side opposite it, and one other side. Unlike most triangle problems, this particular combination of information does not always pin down a single triangle. Depending on the numbers involved, there can be no valid triangle, exactly one, or two genuinely different triangles that both fit the same three given values.
This calculator takes angle A, the side a opposite it, and the other given side b, and works out which of those three outcomes applies, then solves every valid triangle it finds in full, with both remaining angles, the third side and the area.
The formula
The height of the swing formed by side b at angle A is h = b × sin(A). If side a is shorter than that height, it cannot reach the base line at all and no triangle exists. If a equals that height exactly, there is exactly one triangle, with a right angle at B. If angle A is acute and a is longer than the height but shorter than b, two different angles for B both satisfy sin(B) = b × sin(A) / a, one acute and one its obtuse supplement, giving two distinct triangles. In every other case, only one of those two angles keeps the three angles summing to under 180°, so exactly one triangle exists.
| Term | Meaning |
|---|---|
| SSA (side-side-angle) | A triangle described by one angle and two sides, where the angle is not between the two sides, the specific arrangement that produces the ambiguous case. |
| Ambiguous case | The situation where an SSA triangle has two valid solutions rather than one, because two different angles share the same sine value. |
| Swing height | The value b × sin(A), the shortest possible length side a could have while still reaching the base line, used to decide how many triangles exist. |
The inputs explained
| Field | What to enter |
|---|---|
| Angle A (°) | The known angle, given in degrees, between 0° and 180°. |
| Side a (opposite angle A) | The side directly opposite angle A. This is the side whose length decides how many solutions exist. |
| Side b (the other given side) | The other known side, adjacent to angle A rather than opposite it. |
When to use it
Solving a triangle from a bearing and two distances
Surveying and navigation problems often supply an angle and two distances rather than a full set of angles, which is exactly the SSA setup where more than one triangle can fit the same figures.
Checking whether a homework SSA problem has one answer or two
Students are frequently caught out by SSA questions that quietly have two valid solutions. Running the same angle and sides through this calculator shows immediately whether a second triangle needs to be reported.
Confirming a measurement set is even physically possible
If a proposed angle and pair of side lengths cannot form a triangle at all, this calculator flags that directly, which is a faster check than working through the law of sines by hand only to hit an invalid sine value.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How many triangles exist as side b grows, with angle A and side a fixed
A fixed acute angle and a fixed side a, against a range of values for side b.
| Side b | Number of triangles | Triangle 1: Angle B | Triangle 2: Angle B |
|---|---|---|---|
| 3 | 1 | 17.46° | N/A |
| 6 | 2 | 36.87° | 143.13° |
| 8 | 2 | 53.13° | 126.87° |
| 10 | 1 | 90.00° | N/A |
| 12 | 0 | N/A | N/A |
How many triangles exist as angle A increases, with sides a and b fixed
Fixed side lengths, against a range of values for angle A.
Questions
What is the ambiguous case in trigonometry?
It is the SSA (side-side-angle) situation in the law of sines where the given angle and two sides can be satisfied by two different triangles, rather than the single triangle produced by other combinations like ASA or SAS.
Why does law of sines with two angles and a side not have this problem?
Two angles and a side (AAS or ASA) fix the third angle immediately by subtracting from 180°, leaving no choice to make. The ambiguity only appears when the known angle is not between the two known sides, which is unique to the SSA arrangement.
How do I know if my triangle is the ambiguous case?
Check what information you were given: if it is one angle plus the side opposite that angle plus one more side, and the angle is acute, you may have a two-solution situation. Right or obtuse given angles, or angles that are included between the two known sides, do not produce ambiguity.
What happens if I enter values with no valid triangle?
The calculator reports zero triangles and explains why, either because side a is too short to reach the base line, or because an obtuse or right angle A paired with a shorter opposite side makes closing the triangle impossible.
For the two-angle, one-side case where exactly one triangle always exists, see the law of sines calculator. To solve a triangle from three known sides instead, see the triangle calculator.