Triangle Solver Calculator

Background.

The Quanta triangle calculator solves any triangle — right, acute, or obtuse — from any of the three classical congruence inputs: SSS (three sides), SAS (two sides and the included angle), or ASA (two angles and the included side). It returns all six elements of the triangle (three sides and three angles), the area, and the perimeter, with every result derived from first principles using the Law of Sines and the Law of Cosines rather than any approximation. Every triangle has exactly six elements: three side lengths conventionally labelled a, b, and c, and three interior angles labelled A, B, and C, where each angle sits opposite the side of the same letter. Euclidean geometry tells us that any three of those six measurements — provided the three are chosen sensibly — uniquely determine the triangle, a result Euclid recorded in Book I of the Elements as the three congruence theorems still taught in every geometry class today.

SSS (Proposition 8) says three sides fix the triangle. SAS (Proposition 4) says two sides and the angle they enclose fix it. ASA (Proposition 26) says two angles and the side between them fix it. AAS is just ASA with the angle-sum identity applied first, so it collapses to the same solver. Only one classical case, SSA — two sides and a non-included angle — is genuinely ambiguous: depending on the numbers, an SSA input can correspond to zero, one, or two distinct triangles. That is why this calculator deliberately exposes the three unambiguous modes and folds SAS and ASA into the same engine that handles SSS, letting you compare results across modes whenever you have redundant data.

Behind every output is a chain of three classical identities — the angle-sum rule A + B + C = 180°, the Law of Sines a/sin A = b/sin B = c/sin C = 2R (where R is the circumradius), and the Law of Cosines c² = a² + b² − 2ab·cos C — together with one elegant area formula attributed to Hero of Alexandria around 60 CE, K = √(s(s − a)(s − b)(s − c)), where s is the semiperimeter (a + b + c)/2. Whether you arrived here to finish a surveying problem, decode a structural-engineering load triangle, plan a sail-cut, double-check a homework answer, or simply remember the trig you last met in tenth grade, this page walks through every mode the calculator supports, the geometry that justifies it, the ambiguous-case trap that SSA inputs hide, and a fully worked 3-4-5 example you can verify by hand in under a minute.

What is triangle solver calculator?

A triangle is the simplest closed polygon — three vertices, three sides, three angles — and it is the foundational shape of all Euclidean geometry because every other polygon can be triangulated into a finite set of triangles whose properties together determine the whole. Three relationships govern every triangle in the plane. First, the angle-sum identity: A + B + C = π radians, or equivalently 180°. This was proved by Euclid in Book I, Proposition 32 of the Elements and follows immediately from the parallel postulate. Second, the Law of Sines: a / sin A = b / sin B = c / sin C = 2R, where R is the radius of the triangle's circumscribed circle. The ratio is the same for all three side-angle pairs, which is why the Law of Sines is the natural tool whenever you know an angle and its opposite side and want to find another side or another angle. Third, the Law of Cosines: c² = a² + b² − 2ab·cos C, which generalises the Pythagorean theorem to non-right triangles — when C = 90°, cos C = 0 and the formula reduces to c² = a² + b². The Law of Cosines is the natural tool whenever you know all three sides (SSS) and want an angle, or whenever you know two sides and the included angle (SAS) and want the third side. Area can be computed two equivalent ways: directly from two sides and their included angle, K = (1/2)·a·b·sin C, or purely from the three sides via Heron's formula, K = √(s(s − a)(s − b)(s − c)) where s = (a + b + c)/2. Heron's formula, recorded in his Metrica around 60 CE, is remarkable because it lets you compute area without ever explicitly finding an angle or a height — a fact medieval surveyors used routinely. Together these four identities are everything you need to solve any triangle in any mode; the calculator just packages them behind a single interface.

How to use this calculator.

1. 01Pick the mode that matches the data you actually have. SSS if you know all three side lengths. SAS if you know two side lengths plus the angle wedged between them (the included angle, at vertex C in this calculator's convention). ASA if you know two angles plus the side that connects them (the included side, opposite vertex C). If you have AAS — two angles and a non-included side — subtract from 180° to get the third angle, then enter it as ASA.
2. 02Enter the three values your chosen mode requires. In SSS, fill in side a, side b, and side c. In SAS, fill in side a, side b, and angle C. In ASA, fill in angle A, angle B, and side c. Leave the unused fields at their default of 0 — the engine ignores them based on the selected mode.
3. 03Make sure every side is strictly positive and every angle is strictly between 0° and 180°. In ASA mode the two entered angles must additionally sum to less than 180°, because the third angle is computed as 180° − A − B and must itself be positive.
4. 04In SSS mode, check the triangle inequality before trusting the result: every side must be strictly less than the sum of the other two. If you typed 1, 2, 10 the calculator will refuse the geometry — no triangle exists with those side lengths because the longest side exceeds the other two combined.
5. 05Read the primary output (area) and the full breakdown of all three sides and all three angles. Use the area for surface-area, material-cost, or coverage problems, and use the perimeter for fencing, framing, or boundary-length problems.
6. 06If you have SSA data (two sides and a non-included angle), the problem is genuinely ambiguous — depending on the numbers, zero, one, or two distinct triangles satisfy your inputs. Use a dedicated SSA solver or split the case manually; the SSS/SAS/ASA modes here only return unique triangles by design.

The formula.

Three identities do all the work. The Law of Cosines, c² = a² + b² − 2ab·cos C, is the workhorse of SSS and SAS. Derive it by placing vertex C at the origin and aligning side b along the positive x-axis: vertex A sits at (b, 0) and vertex B sits at (a·cos C, a·sin C). The distance between A and B is c, so by the standard distance formula c² = (b − a·cos C)² + (a·sin C)² = b² − 2ab·cos C + a²·cos²C + a²·sin²C = a² + b² − 2ab·cos C, using sin² + cos² = 1. In SSS mode the calculator rearranges that to cos A = (b² + c² − a²) / (2bc) and cos B = (a² + c² − b²) / (2ac), then computes angle C as 180° − A − B. In SAS mode it uses the identity in its original direction to find c, then applies the Law of Sines to recover angle A. The Law of Sines, a/sin A = b/sin B = c/sin C, is the workhorse of ASA. Derive it by dropping a perpendicular h from vertex C onto side c: from the two right triangles created, h = b·sin A and h = a·sin B, so a/sin A = b/sin B; symmetry gives the third ratio. In ASA mode the calculator computes the third angle C = 180° − A − B, then uses a = c·sin A / sin C and b = c·sin B / sin C to recover both unknown sides in a single pair of multiplications. The right tool follows from the data. Use the Law of Cosines whenever you have a complete side-angle-side relationship around one vertex — all three sides (SSS), or two sides and the angle between them (SAS) — because the law involves exactly those four quantities. Use the Law of Sines whenever you have a matched angle-and-opposite-side pair (ASA after computing the third angle, or AAS), because the law's ratios link each angle directly to its opposite side. Area is computed via K = (1/2)·a·b·sin C in every mode (once a, b, and C are known), and the perimeter is simply a + b + c. The calculator also validates geometry: SSS rejects triples violating the triangle inequality (a ≥ b + c, etc.), SAS rejects angles outside (0°, 180°), and ASA rejects angle pairs whose sum is ≥ 180°.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Euclid, Elements, Book I, Proposition 1 — the construction of an equilateral triangle on a given finite straight line, the founding theorem of plane geometry (Heath translation, Cambridge University Press, 1908).
2. [2]Euclid, Elements, Book I, Proposition 4 — the SAS (side-angle-side) congruence theorem: if two triangles share two sides and the included angle, they are congruent.
3. [3]Euclid, Elements, Book I, Proposition 32 — the angle-sum theorem: the interior angles of any triangle sum to two right angles (180°).
4. [4]Euclid, Elements, Book I, Proposition 47 — the Pythagorean theorem, the special case of the Law of Cosines when the included angle is 90°.
5. [5]Berggren, J. L. Episodes in the Mathematics of Medieval Islam, 2nd edition. Springer, 2016 — covers al-Khwarizmi (~825 CE) and the medieval Arabic mathematicians who systematised the Law of Sines and Law of Cosines and transmitted them to medieval Europe.
6. [6]NIST Digital Library of Mathematical Functions, §4 — Elementary Functions (Trigonometric). Reference identities including angle-sum, Law of Sines, Law of Cosines, and Heron's formula.
7. [7]Coxeter, H. S. M. Introduction to Geometry, 2nd edition. Wiley, 1969 — classic reference textbook covering the congruence theorems (SSS, SAS, ASA), the Law of Sines, the Law of Cosines, Heron's formula, and the ambiguous SSA case.
8. [8]Weisstein, Eric W. "Triangle." MathWorld — Wolfram Research. Comprehensive reference for triangle identities, including side-angle formulas, area formulas, and the relationship between the circumradius R and the Law of Sines (a/sin A = 2R).