Compute angles, area, heights and radii of a triangle from its three sides.
Enter the three side lengths. The calc applies law of cosines and Heron's formula.
Law of cosines: cos A = (b² + c² − a²) / (2bc), repeated for B and C. Heron's formula: area = √(s·(s−a)·(s−b)·(s−c)) with s = (a+b+c)/2. The triangle inequality requires every side to be less than the sum of the other two.
The triangle is the simplest planar polygon and the building block of trigonometry, surveying, navigation, and computer graphics. Solving a triangle — finding all six elements (three sides, three angles) from the available subset — is the canonical exercise in classical geometry. The most common case is SSS (three sides given): from the three side lengths, the law of cosines gives all three angles, Heron's formula gives the area, and elementary geometry gives heights, perimeter, inradius and circumradius. This calculator implements the SSS solve plus diagnostics — type classification (right/acute/obtuse, equilateral/isosceles/scalene) — and a scale-correct visual rendering.
Given sides a, b, c (must satisfy the triangle inequality: every side is strictly less than the sum of the other two; if not, no triangle exists):
Enter the three side lengths a, b, c. Any positive numbers work; the units are unspecified (cm, m, in, anything — the area comes out in the squared unit). The calc validates the triangle inequality first; if violated, it returns "–" and a one-line note. Otherwise it returns all six elements (three angles in degrees, three sides as entered), area, perimeter, three altitudes, inradius, circumradius, and a type classification, plus a scale-correct SVG diagram with vertex labels and angle annotations.
Right triangle 3-4-5 (the classic Pythagorean triple).
Equilateral 6-6-6: angles all 60°, area = (√3/4)·6² ≈ 15.59, perimeter 18, heights all (√3/2)·6 ≈ 5.196, inradius √3 ≈ 1.732, circumradius 2√3 ≈ 3.464.
Floating-point precision near degenerate triangles. A triangle with sides 1, 1, 1.999999 is barely valid; the law of cosines gives an angle very close to 180°, and small input errors blow up the computed angles. The calc handles the strict inequality (≤ rejected) but doesn't warn near-degenerate cases.
Triangle-inequality check is strict. Sides 3, 4, 7 fail (3 + 4 = 7, equality); the calc rejects. A "degenerate triangle" (collinear points) has area 0 and is geometrically not a triangle.
Right-angle detection floating-point. The calc detects right angles when |angle − 90°| < 0.01°. A genuinely right triangle entered with rounded sides (e.g., 7-7-9.9 instead of 7-7-9.899...) won't classify as Right; the type label gracefully falls back to Acute / Obtuse.
No SAS, ASA, AAS support. The calc only does SSS (three sides). For SAS (two sides + included angle), AAS (two angles + non-included side), or ASA (two angles + included side), you need the law of sines as a complementary tool. SSA is ambiguous (the "ambiguous case") and can have 0, 1, or 2 valid triangles.
Negative areas are nonsense. Heron's formula can return a complex number for invalid inputs; the calc filters via the triangle-inequality check first.
Coordinate-based input. Some users want to enter three (x, y) vertex coordinates instead of three side lengths. Compute the side lengths first (Euclidean distance) and feed those.
Spherical / hyperbolic triangles. The flat-plane formulas here assume Euclidean geometry. Triangles on a sphere (geodesics on Earth) have angle sums > 180°, and on hyperbolic surfaces < 180°. Different formulas (spherical law of cosines, etc.).
Legal / surveying triangulation. For real surveying you need angle-based methods (theodolite measurements), not just side lengths. The calc is for pure geometry, not practical land surveying.
Equilateral classification edge cases. Sides 5.000, 5.001, 5.000 are almost equilateral but technically isosceles. The calc uses a 0.001 tolerance; tighten or loosen as you need.