SSA Triangle Calculator

What Is an SSA Triangle?

An SSA triangle is defined by two sides and a non-included angle (Side-Side-Angle). This is the ambiguous case of the law of sines: zero, one, or two valid triangles may exist.

SSA Formula

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

Worked Example

Given $a = 31$, $b = 27$, and $\alpha = 46°$, we get $\sin \beta = (27 \times \sin 46°) / 31 \approx 0.629$, so $\beta \approx 38.79°$ and $c \approx 24.12$.

Frequently Asked Questions

Why is SSA called the ambiguous case?

When the known angle is acute and the side opposite it is shorter than the adjacent side, two different triangles can satisfy the given measurements.

When does no SSA triangle exist?

When side a is shorter than the altitude b × sin(α). In that case, side a cannot reach the base to close the triangle.

Does SSA prove triangle congruence?

No. SSA alone does not guarantee congruence because of the ambiguous case. You need additional information or must check both possible solutions.

Which angle should I enter as α?

Enter the angle opposite side a. Side a must be the side across from the known angle.

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What Is an SSA Triangle?

SSA Formula

Worked Example

Frequently Asked Questions

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