Any triangle, from whichever three things you know.

Three laws do all the work

• Law of cosines: c² = a² + b² − 2ab·cos(C). Finds a side from two sides and the included angle, or an angle from three sides.
• Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Finds sides when you know two angles and one side.
• Heron’s formula: area = √(s(s−a)(s−b)(s−c)) where s is half the perimeter. Gives the area from three sides with no angle or height required.

Heron’s formula is the useful one for practical work, because measuring three lengths is easy and measuring a perpendicular height is not.

The triangle inequality

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.

Why SSA is missing

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.

What is Heron’s formula?

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.

Why can I not enter two sides and a non-included angle?

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.

What makes three lengths impossible as a triangle?

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.

How do I find the angles from three sides?

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.