Triangle Calculator

These are the four standard congruence cases for triangles — the four sets of three independent pieces of information that pin down a triangle up to congruence. SSS = three sides; SAS = two sides and the angle between them; ASA = two angles and the side between them; AAS = two angles and a side not between them. Any of these is enough to recover every other side and angle.

The law of cosines generalises the Pythagorean theorem to any triangle: c² = a² + b² − 2ab cos(C), where C is the angle opposite side c. When C = 90° the cosine is zero and you get back c² = a² + b². It's the natural tool for SSS (solve for an angle when all three sides are known) and SAS (solve for the third side when two sides and the included angle are known).

The law of sines says that in any triangle, the ratio of each side to the sine of its opposite angle is the same constant: a / sin A = b / sin B = c / sin C = 2R, where R is the circumradius. It's used to solve ASA and AAS triangles, and it gives the circumradius directly from any side-angle pair.

This calculator uses Heron's formula. With semi-perimeter s = (a + b + c) / 2, the area is √(s(s − a)(s − b)(s − c)). The advantage is that it works from the three sides alone — you don't need an altitude or a particular angle. For triangles where you already know an angle, you can also use ½·a·b·sin(C), which is mathematically equivalent.

SSA (two sides and a non-included angle) is the famous ambiguous case: depending on the values, it can produce zero, one, or two valid triangles. Because the answer isn't unique, this calculator omits SSA — use SAS or AAS if you can reconfigure your knowns. If you're stuck with SSA in a textbook problem, apply the law of sines to find the second angle and check whether the result fits inside (0°, 180°) — it may also produce a second valid solution (180° − that angle).

The inradius r is the radius of the largest circle that fits inside the triangle, tangent to all three sides. It equals area / s, where s is the semi-perimeter. The circumradius R is the radius of the unique circle that passes through all three vertices, and equals a / (2 sin A) — the same constant from the law of sines. Both are useful for inscribed/circumscribed problems and for some advanced geometric identities.

Triangles are classified by angles and by sides. By angles: acute (all angles < 90°), right (one angle = 90°), obtuse (one angle > 90°). By sides: scalene (all sides different), isosceles (two sides equal), equilateral (all three sides equal, which forces all angles = 60°). Every triangle has one label from each list — for example, "acute scalene" or "right isosceles".

This is the triangle angle-sum theorem: in Euclidean geometry, the three interior angles of any triangle sum to 180° (or π radians). The classic proof draws a line through one vertex parallel to the opposite side and uses alternate interior angles. The theorem fails on a curved surface — on a sphere the angles sum to more than 180°; in hyperbolic geometry, less.

The triangle inequality says the sum of any two sides of a triangle is strictly greater than the third side: a + b > c, b + c > a, and a + c > b. If this fails for your three side lengths, no triangle exists with those sides. The calculator will flag impossible side triples — they cannot be drawn without one side overshooting the sum of the other two.

Yes — the SSS/SAS/ASA/AAS framework is exactly what surveyors and navigators use to solve triangulation problems. Field measurements typically give you two angles from a known baseline (ASA) or two sides and an angle (SAS). The calculator gives you the missing side or angle, the area, and the diagram. For high-precision surveying, account for Earth-curvature and atmospheric refraction — flat-earth Euclidean triangle solving is fine for distances under a few kilometres.

The calculator uses IEEE 754 double-precision floating point, accurate to about 15–17 significant figures. For triangles with very obtuse angles or very nearly degenerate shapes (one angle close to 0° or 180°), small input errors can amplify into noticeable output errors — this is a property of the geometry, not the implementation. For critical engineering use, double-check with independent calculation or symbolic algebra.

Yes — the diagram is drawn from the computed coordinates of the three vertices and scaled to fit the SVG viewport. Relative side lengths are accurate; absolute size depends on the viewport dimensions. The labels A, B, C identify the vertices; a, b, c identify the opposite sides, following the standard convention.