SSA Triangle Calculator

How to use this SSA triangle calculator

What this calculator does

This calculator solves an oblique triangle when you know two side lengths and one angle opposite one of those sides – the side-side-angle, or SSA, case. It applies the Law of Sines, tests the supplementary angle, and reports whether the measurements produce no triangle, one triangle, or two different triangles. It then completes each valid triangle by finding the remaining angle, side c, perimeter, and area. The result is a mathematical solution for ideal measurements; it does not account for surveying tolerances, construction error, or uncertainty in measured values.

When to use it

Use it to check trigonometry homework, solve a surveying or navigation triangle, verify a CAD or fabrication sketch, or investigate whether an SSA data set has the ambiguous two-solution case. The underlying relationship is the Law of Sines for non-right triangles.

How to calculate

1. The calculator opens with the complete demonstration values Side a = 31, Side b = 27, and Angle α = 46°. Its result and a validated example workbook are available immediately.
2. In Known angle / formula, select Angle α known – solve β when α is your given angle, or select Angle β known – solve α when β is your given angle. The inactive angle field becomes disabled because it is the angle being solved.
3. Replace Side a and Side b with positive lengths expressed in the same unit. Then enter the active known angle in degrees. Results update as you type.
4. Read Triangle outcome first. Review the Solution 1 cards, then use Valid solution comparison when two triangles are possible.
5. Select Download Excel to export the current inputs and every valid solution. Reset clears the demonstration and all calculated state; it may disable Download Excel until a complete valid input set is entered again.

Input guide

Known angle / formula is required and chooses the side-angle pair used as the known ratio. It accepts one of two fixed options. For example, choose “Angle α known – solve β” for a = 31, b = 27, α = 46°. A common mistake is selecting β as known while typing the given angle into α.

Side a and Side b are required positive decimal lengths up to 1,000,000,000. They must use the same unit, such as meters, feet, or centimeters. Plain decimals and correctly grouped U.S.-style numbers are accepted; decimal commas and scientific notation are rejected to avoid reinterpretation. Increasing the side opposite the unknown angle increases the sine ratio and may change the outcome from one solution to two or to no solution.

Angle α and Angle β accept decimal degrees strictly between 0° and 180°. Only the angle selected as known is required; the other field is disabled and solved by the model. Example: enter 46 in Angle α. Do not enter radians, degree-minute-second notation, or a degree symbol in the input.

Output guide

Triangle outcome and the solution count pill identify zero, one, or two mathematically valid triangles. classification explains why the supplementary candidate is accepted or rejected. Solved opposite angle, Angle γ, and Side c complete Solution 1. Perimeter is a length total; Area is in square units. Sine ratio is dimensionless: a value above 1 means no real angle can satisfy the data. Current equation and its interpretation show the exact Law of Sines step used.

The Valid solution comparison table lists Solution, Angle α, Angle β, Angle γ, Side a, Side b, Side c, Perimeter, and Area. Every angle row should total 180°. When two rows appear, both satisfy the same three original inputs but have different remaining geometry; neither row is an approximation of the other.

Worked example

With Side a = 31, Side b = 27, and Angle α = 46°, the calculator evaluates sin(β) = 27 × sin(46°) ÷ 31 = 0.626563. The principal inverse-sine result is β = 38.794°. Its supplement, 141.206°, would make α + β exceed 180°, so it is rejected. The remaining angle is γ = 180° – 46° – 38.794° = 95.206°. The Law of Sines gives Side c = 42.917 units, producing a perimeter of 100.917 units and an area of 416.774 square units. These figures match the first-open cards, table, and workbook checkpoints.

Learn more

OpenStax publishes the standard Law of Sines equation set, while Mathematics LibreTexts explains why the inverse sine can create an ambiguous SSA case.