How this calculation works
Choose a set of measurements that determines one triangle, then follow the matching sine or cosine rule. The solver returns every side and angle, plus area and perimeter. Worked steps make the side-angle correspondence explicit, while strict validity checks reject measurements that cannot form a triangle.
Worked example
This example uses illustrative inputs. Your current result is shown above.
For SSS with a = 3, b = 4 and c = 5, s = 6, area = √[6×3×2×1] = 6 square units and perimeter = 12 units. The opposite angles are A ≈ 36.86989765°, B ≈ 53.13010235° and C = 90°. For SAS with b = 3, c = 4 and included A = 90°, the missing opposite side is a = 5.
Assumptions and limits
- The triangle is a flat Euclidean triangle. Lowercase side names are opposite the matching uppercase angle names.
- SSS uses a, b and c. SAS uses b, c and their included angle A. ASA uses c between known angles A/B. AAS uses a opposite known angle A, plus angle B.
- Input sides are positive and between 10⁻¹² and 10¹² in one consistent unit. Input angles are in degrees from 0.00000001° to 179.99999999°; two known angles must sum to less than 180°.
- SSS must satisfy strict triangle inequalities. Degenerate straight-line cases, SSA ambiguity and angles-only AAA are not solved.
- Calculation retains 80 significant digits before numeric conversion. Results show up to 12 significant digits; near-degenerate triangles remain sensitive to rounded inputs. Unsupported extreme cases produce a field error, not a fabricated triangle.
- Solved lengths can exceed the input-side bounds. Area is in squared length units; the diagram is a visual guide and must not be used as a measurement.
Questions people ask
Which measurements should I enter for each mode?
SSS: all three sides a, b and c. SAS: sides b and c, with their included angle A. ASA: side c and the angles A/B at its ends. AAS: side a, its opposite angle A, and angle B. Rename the vertices of your triangle consistently to match these conventions.
Why is SSA not an option?
Two sides and a non-included angle can give zero, one or two triangles. This solver intentionally supports only the uniquely determined SSS, SAS, ASA and AAS cases; it does not silently choose one of two possible SSA answers.
Can I use centimetres, metres or inches?
Yes, as long as every entered side uses the same unit. The result keeps that unit for lengths and perimeter, and its square for area. Angles must always be entered in degrees, not radians.
Why are some nearly flat triangles rejected?
The sum of any two sides must be strictly greater than the third. Exact equality forms a line, not a triangle. Even when mathematically valid, exceptionally extreme inputs can round a computed angle to 0° or 180°; these produce an error rather than misleading geometry.