One fixed side–angle mapping for every method
In this triangle angle calculator, lowercase a is always opposite uppercase A, b is opposite B, and c is opposite C. This convention matters in the sine law a/sin A = b/sin B = c/sin C and the cosine law a² = b² + c² − 2bc cos A. Automatic mode examines the supplied values, but the result still identifies which family was actually solved.
SSS supplies three sides; SAS supplies two sides around their included angle; ASA or AAS supplies two angles and a side; SSA supplies an angle, its opposite side, and another side. AAA determines proportions only, not a unique size. Solve commits the current inputs so screen readers are not interrupted by a new result after every keystroke.
How the triangle solver selects and checks a method
- Keep each side beside its opposite capital-letter angle. Enter only the measurements that define the intended problem.
- Choose Auto to detect SSS, SAS, ASA/AAS, or SSA, or choose a method to document your intended input family.
- Select Solve triangle. All sides must be positive, each angle must be strictly between 0° and 180°, and a completed triangle must satisfy the strict side inequalities.
- If SSA creates two candidates, compare Solution 1 and Solution 2 as separate valid triangles; the calculator does not prefer one.
- Use the displayed method steps, unrounded internal calculation, and angle-sum check before copying or sharing the result.
Formula choices, clamping, and tolerances
- SSS uses the cosine law after checking a + b > c, a + c > b, and b + c > a. Inputs to acos are clamped to [−1, 1] only to absorb floating-point spillover.
- SAS first computes the opposite side with x² + y² − 2xy cos θ, then solves the completed SSS triangle. The supplied angle must be the included angle.
- ASA/AAS obtains the third angle from 180° and applies one sine-law scale. At least one side is required because angles alone leave infinitely many similar triangles.
- SSA tests asin(x) and its supplement 180° − asin(x). Each candidate must leave a positive third angle and pass side validation; floating-point duplicates are removed.
- Calculations retain full precision. Display rounding can make printed angles appear a final decimal short of or beyond 180°, so the internal tolerance check is reported separately.
Worked unique and ambiguous examples
- Unique SSS example: a = 3, b = 4, c = 5 gives A ≈ 36.870°, B ≈ 53.130°, and C = 90°. The area is 6 square units and the perimeter is 12 units.
- Unique SAS example: b = 5, c = 8, and included A = 60° gives a = 7 by the cosine law. The remaining angles are approximately B = 38.213° and C = 81.787°.
- Ambiguous SSA example: A = 30°, a = 10, and b = 12 gives sin B = 0.6. Both B ≈ 36.870° and B ≈ 143.130° leave a positive C, producing two different valid triangles.
- No-solution SSA example: A = 30°, a = 4, and b = 10 would require sin B = 1.25. No real angle has that sine, so no triangle is shown.
Why the SSA case can have two triangles
The sine function has the same positive value at an acute angle θ and its obtuse supplement 180° − θ. With a known opposite pair, either candidate may fit the remaining side. A right or obtuse known angle often eliminates the supplementary candidate because the angle sum would reach or exceed 180°.
- Generate the principal inverse-sine candidate.
- Test its supplement independently.
- Require a positive final angle and valid sides.
- Show zero, one, or two solutions explicitly.
Triangle inequality and scale
For positive a, b, and c, every pair must total more than the remaining side. Equality is a collapsed segment with zero area, not a triangle. AAA is also excluded as a size solution: 40°, 60°, and 80° describe a family of similar triangles until one length sets the scale.
Degenerate shapes and measurement limits
A nearly straight triangle may satisfy strict mathematical inequalities while remaining extremely sensitive to input rounding. The tool warns when a side gap or angle is tiny. Diagrams are scaled illustrations, not engineering drawings, and units remain generic unless the user keeps all side entries in the same unit. The solver cannot infer whether rounded field measurements describe an exact intended triangle.
All triangle calculations run locally. A share URL contains only the selected mode and numeric fields.