Three sides, two sides and the angle between them, or two angles and a side — each determines the triangle completely.
This page's own default — sides 3, 4, 5 (SSS) — isn't the only way to describe that triangle. Entering it as SAS instead (side 3, included angle 90°, side 4) solves the missing side as exactly 5.000000. Entering it as ASA instead (angle 36.8699°, side 5, angle 53.1301°) solves the other two sides as 3.000000 and 4.000000. All three modes are different descriptions of one triangle, and the calculator reconstructs the identical shape from any of them.
Heron's formula gives the 3-4-5 triangle's area as exactly 6 from the three sides alone. The more familiar ½ × base × height formula, using the two legs as base and height with the law-of-cosines-derived 90° angle between them, gives ½ × 3 × 4 × sin(90°) = 6 as well. Two genuinely different routes to area — one needing only side lengths, one needing an angle — landing on the same number is what confirms the angle computed from the sides is actually correct, not just a plausible-looking one.
Side a = 7, angle A = 30°, side b = 10 — a side, a non-included angle, and a second side — genuinely describes two different triangles. The law of sines gives sin(B) = 10 × sin(30°) ÷ 7 ≈ 0.7143, and both B ≈ 45.58° and B ≈ 134.42° satisfy that equation while keeping every angle positive and under 180° combined with A. One resolves to a third side near 13.56 with an obtuse 104.42° angle at C; the other resolves to a third side near 3.76 with an acute 15.58° angle at C. Both are entirely valid triangles built from the exact same three starting numbers, which is precisely why this input combination is left out rather than silently picking one for you. For the special case where one angle is fixed at exactly 90°, the Right Triangle Calculator and Pythagorean Theorem Calculator both sidestep this ambiguity entirely, since a right angle plus one side and one more piece of information is always enough on its own. For the area of shapes that are not triangles at all, the Area Calculator covers the rest of the common 2D shapes.
Heron’s formula is the useful one for practical work, because measuring three lengths is easy and measuring a perpendicular height is not.
Any two sides must sum to more than the third. Sides of 2, 3 and 9 cannot form a triangle: the short sides laid end to end still fall short of the long one. This calculator checks the condition and tells you rather than returning an imaginary area.
The check matters because Heron’s formula fails silently otherwise — it produces the square root of a negative number, which some implementations quietly render as an error or a zero.
You may notice there is no option for two sides and a non-included angle. That is the ambiguous case: the information can describe two genuinely different triangles, one acute and one obtuse, and no calculator can tell you which you meant. Rather than silently picking one, it is left out.
SSS, SAS and ASA are all unambiguous. A triangle with sides 3, 4 and 5 has angles of 36.87°, 53.13° and exactly 90°, and an area of 6 — the familiar right triangle, arrived at without assuming it was one.
A way to find a triangle’s area from its three side lengths alone: area = √(s(s−a)(s−b)(s−c)), where s is half the perimeter. It needs no angles and no perpendicular height, which makes it the practical choice when you can measure sides but not heights.
Because that combination is ambiguous — it can describe two different valid triangles. Rather than guess which one you meant, the calculator offers only the three combinations that give a single answer.
If any two sides do not sum to more than the third. Sides of 2, 3 and 9 fail because 2 + 3 is less than 9, so the two short sides can never meet.
Rearrange the law of cosines: cos(C) = (a² + b² − c²) ÷ 2ab, then take the inverse cosine. Do it for two angles and subtract from 180° for the third.
A triangle with one angle fixed at exactly 90° is a special case worth its own tool — the Right Triangle Calculator skips the law of cosines entirely and solves directly from simple trigonometric ratios instead.
For right triangles specifically, the Right Triangle Calculator and the Pythagorean Theorem Calculator are more direct. For area from base and height rather than three sides, use the Area Calculator.