Triangle length calculator
Solve a triangle from three known measurements, including at least one angle and one side. The calculator handles ASA, AAS, SAS, and the ambiguous SSA case.
Known measurements
Enter any valid set of three measurements. Keep unknown fields empty. Angles are in degrees; side lengths may use any consistent unit.
Live solution
One unique triangle solved with the law of cosines.
Solution details
| Solution | Method | Angle α | Angle β | Angle γ | Side a | Side b | Side c |
|---|---|---|---|---|---|---|---|
| 1 | SAS | 36.8699° | 53.1301° | 90.0000° | 3.0000 | 4.0000 | 5.0000 |
The startup values form a unique 3 – 4 – 5 right triangle. SSA inputs can produce two valid rows because the known measurements may describe two different triangles.
How to use this triangle length calculator
What this calculator does. This tool solves the side lengths and interior angles of a plane triangle when you supply a sufficient set of measurements. It is designed for the common ASA, AAS, SAS, and SSA arrangements and uses the angle-sum rule, the law of sines, and the law of cosines. It does not infer scale from a drawing, verify surveying instruments, or account for curved surfaces. Side values are unit-neutral, so every side must use the same unit.
When to use it. Use it to find a missing span from two measured sides and the included angle, check a trigonometry exercise, solve a layout triangle when two angles and one baseline are known, or examine whether an SSA measurement set has one or two possible solutions. The OpenStax explanation of the law of sines provides a detailed derivation and examples of ASA, AAS, and SSA triangles.
How to calculate. The calculator opens with a ready-to-use demonstration: Angle γ = 90, Side a = 3, and Side b = 4. Its first result and validated Excel workbook are immediately available. To use your own measurements:
- Replace the demonstration values with exactly three known measurements. Keep unknown fields empty. Include at least one angle and at least one side.
- Match opposite pairs correctly: Angle α is opposite Side a, Angle β is opposite Side b, and Angle γ is opposite Side c.
- Read the live primary result, the six solved measurements, and the solution table. If an SSA set has two valid triangles, both appear as separate rows.
- Select Download Excel to export the current inputs and canonical results as a real workbook. Select Reset to clear the demonstration and all calculated content; export remains unavailable until a complete valid set is entered again.
Input guide. Angle α, Angle β, and Angle γ accept required or optional positive decimal degree values, depending on which three measurements you know. Each angle must be greater than 0 and less than 180, and known angles must leave room for the remaining positive angle. A realistic entry is 35.5. Increasing an included angle usually increases its opposite side when the adjacent sides stay fixed. Do not enter degree symbols, commas, scientific notation, or radians. Side a, Side b, and Side c accept positive decimal lengths up to 1e150 in one consistent unit. A realistic entry is 12.75. A side is always paired with the angle carrying the same letter. Do not mix centimeters and inches or assume the letters identify horizontal and vertical directions.
Output guide. Known values counts nonempty valid input fields. Solutions reports how many triangles match the measurements. Method identifies ASA/AAS, SAS, or SSA; the Angle sum is the exact triangle identity of 180 degrees. Calculated side or Calculated angle highlights the first missing quantity; when several quantities were unknown it identifies the first missing side, then the first missing angle. The six result cards show the solved Angle α, Angle β, Angle γ, Side a, Side b, and Side c for the first valid solution. The table repeats those values for every solution. A zero, negative, nonfinite, or inconsistent result is never displayed; the calculator instead explains which inputs need correction.
Worked example. The startup set supplies Side a = 3, Side b = 4, and the included Angle γ = 90°. The law of cosines gives c = √(a² + b² – 2ab cos γ). Because cos 90° = 0, c = √(3² + 4²) = √25 = 5. The remaining angles are α = 36.8699° and β = 53.1301°, so the three angles total 180.0000°. These values match the first rendered result, the solution table, and the startup workbook.
Learn more. For the SAS and side-to-angle formulas used here, review the OpenStax law of cosines chapter. It explains why the cosine rule generalizes the Pythagorean theorem and how three sides determine the angles.
How the formulas work
The angle-sum identity supplies a third angle whenever two are known. Once one opposite side-angle pair is available, the law of sines scales the other sides: a/sin α = b/sin β = c/sin γ. With two sides and their included angle, the law of cosines calculates the opposite side before the remaining angles are recovered. These steps preserve full internal precision and round only the displayed values to four decimal places.
Understanding the ambiguous SSA case
SSA means two sides and an angle opposite one of those sides are known. The inverse sine can return an acute angle while its supplementary obtuse angle has the same sine. Depending on the measurements, neither, one, or both choices may produce a valid positive third angle. This calculator tests both possibilities and lists every distinct triangle that matches the supplied values. OpenStax's discussion of possible SSA outcomes illustrates why the case needs this extra check.
Common mistakes
- Entering fewer than three known measurements or entering only angles. Angles determine shape but not absolute side lengths.
- Pairing Angle α with the wrong opposite side. The letter pairing is part of every sine and cosine formula.
- Supplying four or more measurements that do not describe the same triangle. Extra values are accepted only when they agree with a valid solution.
- Mixing units among side fields or typing formatted values such as “1,000”, “5 cm”, or “1e3”. Enter plain decimal numbers and keep one unit throughout.