Triangle Calculator
Solve a triangle from three known measurements. See all sides, angles, area and heights, and compare both possible SSA shapes on one scale.
Enter three known sides or angles. Leave the unknowns blank. See the complete triangle — and both shapes when two answers are possible.
Lowercase a, b, c are sides; uppercase A, B, C are the opposite angles. Use the same length unit for every side.
SSS · a, b, c known
One triangle
- Area (square length units)
- 6
- Perimeter (length units)
- 12
Shape uses the calculated proportions. All measurements are listed below.
Side a = 3 is opposite angle A = 36.8699°.
- Side a
- 3
- Angle A
- 36.8699°
- Side b
- 4
- Angle B
- 53.1301°
- Side c
- 5
- Angle C
- 90°
Heights perpendicular to each side
- Height to side a
- 4
- Height to side b
- 3
- Height to side c
- 2.4
Each height is 2 × area ÷ its side length. For an obtuse triangle, the perpendicular foot can lie outside the side segment.
Results use 6 significant digits. The drawing preserves proportions; it is not a drafting template or a statement of measurement precision.
Why can there be two answers?
With two sides and a non-included angle (SSA), the law of sines can give two angles: B and 180° − B. Keep only those that leave a positive third angle. The two valid triangles share your known side lengths and angle, but have different remaining sides and areas.
For three sides (SSS), use the law of cosines. Two sides and their included angle (SAS) also determine one triangle. With two angles and one side (ASA or AAS), the third angle is 180° minus the other two; the law of sines supplies the missing sides. Three angles alone determine shape, not size.
Formulas and supported precision
Sine law: a / sin A = b / sin B = c / sin C. Cosine law: c² = a² + b² − 2ab cos C. Area = ½ab sin C, or Heron's formula with semiperimeter s = (a + b + c) / 2. The calculation uses normalized sides and stable rearrangements of area and half-angle formulas.
Enter positive decimals with a dot, or scientific notation, with at most 15 significant digits. Sides: 1e-100 to 1e100. Angles: 0.00000001° to 179.99999999°. Each side must be strictly shorter than the other two combined. Near-degenerate shapes with a normalized inequality margin below 1e-12 or angles below 1e-8° are outside reliable resolution. Generic SSA sine ratios within 1e-12 of 1 are also reported as a precision limit rather than a guessed solution count. An exact 30° tangent with a = b/2 is supported.
Sources: OpenStax: law of sines and ambiguous SSA; OpenStax: cosine law and Heron's formula; Kahan: stable area and angle calculations.
