Free Triangle Calculator – Solve Every Case, Including Both Ambiguous Triangles

How Triangle Solving Works and Why the Ambiguous Case Matters

A triangle is fully determined by any three of its six measurements — three sides a, b, c and three angles A, B, C — provided at least one of the three is a side. Give this solver three values and it returns the other three plus the area, perimeter, all three altitudes, all three medians, the inradius, the circumradius and the triangle's classification. Students checking geometry homework, carpenters cutting a roof brace, surveyors closing a traverse and machinists laying out a fixture all need the same thing: the missing measurements and enough working to trust them. The solver names which of the five standard cases it recognised — SSS, SAS, ASA, AAS or SSA — because the case decides which law applies and how many answers exist. Two relations do the heavy lifting. The law of cosines, as stated on Wolfram MathWorld, gives c² = a² + b² − 2ab·cos C, where a and b are two sides and C is the angle between them; rearranged as cos A = (b² + c² − a²) / (2bc), it turns three known sides into three angles. The law of sines says a / sin A = b / sin B = c / sin C, which pairs each side with the angle opposite it and lets you scale a triangle once one side-angle pair is known. Area comes from Heron's formula √(s(s−a)(s−b)(s−c)), where s = (a + b + c) / 2 is the semi-perimeter, or from ½ab·sin C when you have two sides and the included angle. Medians use m_a = ½√(2b² + 2c² − a²), altitudes use h_a = 2·Area / a, the inradius is Area / s and the circumradius is a / (2·sin A). Enter a = 3, b = 4, c = 5 and the law of cosines returns A = 36.8699°, B = 53.1301° and C = 90°, with Heron giving an area of exactly 6, an inradius of 1 and a circumradius of 2.5. Now try the case that separates a careful solver from a careless one: a = 7, b = 10, angle A = 30°. The law of sines gives sin B = 10 × sin 30° / 7 = 0.714286, and arcsin returns 45.5847° — but 180° − 45.5847° = 134.4153° has the identical sine. Both keep the angle sum under 180°, so **two real triangles exist**. The first has B = 45.5847°, C = 104.4153°, c = 13.5592 and area 33.8981. The second has B = 134.4153°, C = 15.5847°, c = 3.7613 and area 9.4032. Same three inputs, two completely different triangles, and this tool shows you both. That second triangle is not a curiosity. A student handed an SSA problem is usually being tested on exactly this: recognising zero, one or two solutions. A calculator that reports only the acute branch will mark half those problems wrong without warning. Outside the classroom, a surveyor working from two measured distances and one bearing needs to know that a second position fits the data equally well before committing to a boundary. A roof framer given a rafter length, a rise and one pitch angle faces the same fork. And anyone verifying a CAD constraint benefits from seeing that the geometry is under-constrained rather than discovering it later. When the second branch pushes the angle sum past 180°, this solver says so explicitly and reports a single triangle — as with 12.4, 7.6 and 125°, where the alternative angle of 149.9° cannot coexist with 125°. A sanity check worth running first: the triangle inequality requires the sum of any two sides to exceed the third. Sides of 1, 2 and 5 fail it, and this tool refuses them by name — "1 + 2 = 3 is not greater than 5" — instead of returning a confident wrong number. Three angles alone are rejected too, since they fix the shape but not the size. In SSA, when the required sine exceeds 1, the short side cannot reach, and the tool says so rather than printing NaN. Intermediates keep full precision and rounding happens only on display, so a right angle reads as exactly 90°. Sides are unit-agnostic: use one unit throughout and areas come out squared. Nothing you type is uploaded.

Triangle Calculator Formula & Method

🔥 Trending Tools