Free SSA Triangle Calculator

The SSA Triangle Calculator is a free online tool designed to handle side-side-angle (SSA) triangle problems using the law of sines. Because SSA configurations do not always produce a unique triangle—they are known as the ambiguous case of the law of sines—this tool also serves as an ambiguous case calculator and triangle ambiguity solver. With it, you can verify whether a valid triangle exists, assess SSA triangle congruence, and determine if the given data leads to one, two, or zero possible solutions.

Law of Sines Applied to SSA

For any triangle with sides $a$, $b$, $c$ and opposite angles $\alpha$, $\beta$, $\gamma$, the law of sines states:

$$\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma}$$

In an SSA scenario, you are provided with two sides and a non-included angle—for example, $a$, $b$, and $\alpha$. Rearranging the law of sines gives the expression for the unknown angle $\beta$:

$$\sin\beta = \frac{b \sin\alpha}{a}$$

Resolving the Ambiguity

The computed value of $\sin\beta$ determines how many triangles satisfy the input:

• No solution: If $\frac{b \sin\alpha}{a} > 1$, the triangle cannot exist.
• One solution: If $\frac{b \sin\alpha}{a} = 1$, or if the supplementary angle ($180^\circ - \beta$) leads to a sum exceeding $180^\circ$ when added to the known angle, only one triangle is valid.
• Two solutions: If $\frac{b \sin\alpha}{a} < 1$ and both $\beta$ and $180^\circ - \beta$ produce angle sums below $180^\circ$, two distinct triangles are possible.

This analysis is the heart of the law of sines ambiguous case, and the calculator performs it instantly to provide a complete picture of the triangle’s status.

Example: Single-Solution SSA Triangle

Take a concrete case: $A = 46^\circ$, $a = 31$, $b = 27$. Applying the side side angle calculator:

$$\sin B = \frac{27 \cdot \sin 46^\circ}{31} \approx 0.6267$$

Thus $B \approx 38.794^\circ$. The supplementary angle $180^\circ - 38.794^\circ = 141.206^\circ$, added to the given $46^\circ$, equals $187.206^\circ$, which exceeds $180^\circ$. Therefore, only one triangle exists, and the calculator confirms congruence with a unique solution.

This example shows how the SSA triangle solver online quickly resolves potential ambiguities and gives a definitive answer.

How to Use the SSA Triangle Calculator

To use this tool, simply input your known side lengths and the non-included angle. The calculator automatically selects the appropriate law-of-sines ratio, performs the ambiguity test, and outputs all unknown angles and side lengths, along with the number of valid triangles. Whether you are a student learning about triangle ambiguity or a professional needing a fast solution, this side side angle calculator provides reliable results in seconds.

Important Note on Congruence

SSA alone is not a standard congruence criterion (unlike SAS, SSS, or ASA). However, when exactly one triangle fits the given data, the SSA triangle can be considered congruent under those specific conditions. The ambiguous case calculator helps identify these cases, distinguishing them from situations with multiple or no solutions.