What makes SSA the "ambiguous case"?

Unlike SAS, ASA, or SSS — which always yield exactly one triangle — the SSA configuration can produce 0, 1, or 2 triangles because the angle is not between the two known sides.

When does SSA give zero triangles?

When sin B = b sin A / a > 1, which means side a is too short to reach the opposite side at the given angle.

When does SSA give exactly one triangle?

When B = 90° (right triangle), when a ≥ b (angle A is opposite the longer side), or when the second candidate B₂ would make the angle sum exceed 180°.

When does SSA give two triangles?

When both B₁ and B₂ = 180° − B₁ each produce a valid angle C > 0° (i.e., A + B < 180° for both candidates).

Which law is used to solve SSA?

The Law of Sines: a/sin A = b/sin B = c/sin C.

Can SSA ever produce a right triangle?

Yes — if sin B = 1, then B = 90° and there is exactly one right-triangle solution.

SSA Triangle Calculator (Ambiguous Case)

Solve the ambiguous SSA triangle case — enter two sides and a non-included angle to find 0, 1, or 2 possible triangles. Shows all angles, sides, area, and perimeter for each solution.

Input — SSA (Ambiguous Case)

Side a (opposite angle A)

Side b

Angle A

Angle Unit

Results

Enter side a, side b, and angle A to solve the triangle.

SSA Reference Cases

a	b	A (°)	Solutions	sin B	Case
3	5	20	2	0.5700	Ambiguous
5	8	40	2	1.0285	Ambiguous
8	12	30	2	0.7500	Ambiguous
10	7	50	1	0.5362	Unique
15	10	60	1	0.5774	Unique
7	7	45	1	0.7071	Unique
1	2	100	0	1.9696	No triangle
2	5	80	0	2.4620	No triangle

Planning notes, formulas, and examples

About the SSA Triangle Calculator (Ambiguous Case)

The SSA Triangle Calculator solves the famously ambiguous SSA (side-side-angle) case in triangle geometry. When you are given two sides and an angle that is not between them, there may be zero, one, or two valid triangles. This calculator determines exactly how many solutions exist and computes all properties for each one.

The SSA configuration — also called the ambiguous case of the Law of Sines — is one of the trickiest setups in trigonometry. Given side a (opposite the known angle A), side b, and angle A, the calculator uses the Law of Sines to find angle B: sin B = b sin A / a. If the resulting value exceeds 1, no triangle exists. If exactly 1, there is one right-triangle solution. Otherwise, two candidate values of B are possible (B and 180° − B), and each must be checked to ensure the angles sum to less than 180°.

For each valid solution the calculator displays all three angles, all three sides (using the Law of Sines to find the third side), the area (½ ab sin C), the perimeter, the triangle type (acute / right / obtuse), and key radii (inradius, circumradius). Visual bars compare the two solutions side by side when applicable. The reference table below lists classic SSA configurations and their solution counts. This calculator is indispensable for trigonometry students, surveyors, and anyone working with indirect triangle measurements.

When This Page Helps

SSA problems are where many otherwise solid triangle workflows break down, because the same measurements can produce two different triangles or none at all. This calculator removes that guesswork by checking the ambiguous case automatically and showing every valid solution side by side. It is useful for trigonometry homework, surveying layouts, navigation problems, and any situation where you need to know whether a single set of measurements determines a unique triangle.

How to Use the Inputs

Enter side a — the side opposite the known angle A.
Enter side b — the other known side.
Enter angle A in degrees — the angle opposite side a.
The calculator determines how many solutions exist (0, 1, or 2).
View full triangle properties for each solution.
Use preset buttons to explore famous SSA configurations.
Check the reference table for typical ambiguous-case scenarios.

Formula used


<p><strong>Law of Sines:</strong> sin B = b · sin A / a</p>
<p>If sin B > 1 → no solution.</p>
<p>If sin B = 1 → one right-triangle solution (B = 90°).</p>
<p>Otherwise B₁ = arcsin(sin B) and B₂ = 180° − B₁ are both candidates.</p>
<p>For each valid B: C = 180° − A − B, c = a · sin C / sin A.</p>
<p><strong>Area:</strong> ½ a b sin C</p>

Example Calculation

Result: 2 valid triangles: one with c ≈ 15.69 and one with c ≈ 5.11.

Enter side a = 8, side b = 12, and angle A = 30°.<ul><li>sin B = (12 × sin 30°) / 8 = 0.75</li><li>First candidate: B ≈ 48.59°, so C ≈ 101.41° and c ≈ 15.69</li><li>Second candidate: B ≈ 131.41°, so C ≈ 18.59° and c ≈ 5.11</li></ul>Because both candidate angles keep the angle sum below 180°, the SSA data creates two different triangles. The calculator reports both solutions instead of forcing you to test them by hand.

Tips & Best Practices

The ambiguous case only arises when the known angle is acute and a < b.
If A ≥ 90° and a ≤ b, there is no solution; if a > b, exactly one solution.
Always check that each candidate angle B keeps the angle sum under 180°.
A common student mistake is forgetting the second solution B₂ = 180° − B₁.
If both solutions exist, one triangle is acute and the other obtuse.

Why SSA Can Produce Two Different Triangles

SSA means you know two sides and a non-included angle. That last detail is what causes trouble. Unlike SSS or SAS, the known angle does not lock the triangle into one shape. If the side opposite the known angle is too short, no triangle closes. If it fits exactly, you get one right triangle. If it is long enough to reach the opposite side in two different ways, you get two valid triangles.

This is why the SSA case is called ambiguous. The same measurements can describe one acute triangle and one obtuse triangle, both matching the original inputs. A reliable solver has to test the second arcsine branch instead of assuming the first answer is the only one.

A Clean Decision Process For The Ambiguous Case

Start with the Law of Sines: sin B = b sin A / a. If that value is greater than 1, no triangle exists. If it equals 1, angle B is 90 degrees and there is exactly one solution. If it is between 0 and 1, compute B₁ = arcsin(sin B) and also test B₂ = 180° − B₁.

The final check is whether A + B stays below 180 degrees for each candidate. Any candidate that leaves a positive angle C is a real solution. This calculator handles those checks automatically and then computes the missing side, area, perimeter, and radii for each surviving triangle.

Reading The Two-Solution Example

With a = 8, b = 12, and A = 30°, the calculator finds sin B = 0.75. That gives B ≈ 48.59° or B ≈ 131.41°. Both work, so two different triangles appear. One has a long third side of about 15.69, while the other has a much shorter third side of about 5.11.

That difference is the whole point of the ambiguous case: identical input data can lead to noticeably different geometry. In surveying or navigation, that usually means you need one more measurement to decide which triangle matches reality.

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

Unlike SAS, ASA, or SSS — which always yield exactly one triangle — the SSA configuration can produce 0, 1, or 2 triangles because the angle is not between the two known sides.