Triangle Solver

What does this triangle solver calculator compute?

A Euclidean triangle is determined by SSS, SAS, ASA, or AAS. SSA may yield zero, one, or two triangles (ambiguous case).

The published formula or method is law of sines / law of cosines. The following references support the stated mathematics or method:

• Gilbert Strang, Calculus — Chapter 1: Introduction to Calculus (MIT OpenCourseWare): Right-triangle definitions of sine, cosine and tangent, the sine/cosine identities, and the law-of-cosines and parallelogram exercises 31–32. Chord length follows by bisecting the radii triangle. Regular-polygon apothems and areas follow by splitting the polygon into equal triangles. The law of sines follows by equating a triangle’s altitude computed from either adjacent side; SSA can give two triangles. These applications are derived from the stated relations. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

SSA can have two valid triangles. Retain every returned solution and distinguish it from no-solution or degenerate cases.

The worked example uses SSS right triangle with c² = 3² + 4². Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.geometry.triangle v1.1.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical policy and complete solutions

Use one consistent length unit for a, b, c; area is in its square and angles are in degrees. Side a is opposite A, b opposite B, and c opposite C. SSS uses a,b,c; SAS uses a,b,C; ASA uses A,B,c; AAS uses A,B,a; SSA uses a,A,b. An SSA request with no triangle returns INVALID_INPUT. Successful SSA requests retain every solution in order: acute B first, then supplementary B. The top-level values repeat solution1; JSON and CSV retain the complete solutions array and all perimeters.

Inputs are strict decimal numbers or decimal strings of at most1024 characters. A/B/C are aliases of A_deg/B_deg/C_deg and must agree when both are supplied. Decimal text is normalized to binary64. Nonzero text converted to zero is NUMERIC_UNDERFLOW; nonzero subnormal inputs and nonnormal positive outputs reject with NUMERIC_RANGE. All positive inputs and reported values must be finite normal binary64 (at least2^-1022 in magnitude); degree-to-radian conversions must also remain normal. Rescale lengths on NUMERIC_RANGE. Explicit empty/null fields are invalid; URL defaults apply only when a field is absent.

SSS triangle inequalities use exact dyadic predicates on normalized binary64 inputs, without a fixed absolute length epsilon. Its area uses the sorted Kahan product and angles use scaled atan2. SAS uses a stable half-angle identity; ASA/AAS use the sine law; SSA uses stable quadratic roots. No small positive result is clamped to zero, and two near-coincident valid SSA solutions are not merged.

NUMERIC_CONDITION is an explicit accuracy boundary. For SSA angles other than exact sine identities at30,90,150 degrees, the relative height discriminant abs(a²-h²)/(a²+h²), h=b sin(A), must exceed1e-8; closer cases require higher precision. For derived sides in SAS/ASA/AAS/SSA, every rounded triangle-inequality gap must exceed64 times Number.EPSILON times the shortest side. A rounded endpoint angle or unresolved side geometry is rejected. Raw-side SSS uses exact inequalities without the derived-side guard.

Near-flat and SSA tangent cases are sensitive to input rounding; even a decimal-to-binary conversion can change the solution count. Typical arithmetic checks use relative5e-10 against independent750-digit values of the stated model, separately checking exact status/count, numeric types and full candidate lists. This is not a bound on input uncertainty or a symbolic exactness guarantee. Figures use equal scale and explicitly omit aspect ratios or candidate separations too small to display faithfully. Read every numerical solution even when no figure can be shown.

Kahan’s author-hosted lecture notes explain numerical pitfalls in needle-like triangles and SSA conditioning. They support algorithmic reasoning, not implementation verification or endorsement.

Interactive solving runs locally. Shared URLs send parameters to the server and may be server-rendered; REST/MCP requests send inputs. Local calculation history is stored in this browser. Analytics and deployment processing are separate data paths; no zero-log, retention-period or fixed-region guarantee is implied. A shared calculation_version identifies the requested version and displays a mismatch notice; it is not a preserved historical execution environment.