Three measurements are usually enough to pin down an entire triangle, except in one specific case where they genuinely aren’t, and most calculators quietly ignore that.
Free math tool
Triangle Solver
Solve any triangle from three known measurements: every side, every angle, and the area.
Two valid triangles fit these measurements (the SSA ambiguous case). Both are shown below; the diagram shows the first.
Runs entirely in your browser. Nothing you enter is sent anywhere.
Which law solves which case
The law of cosines (a² = b² + c² − 2bc·cos A) handles SSS and SAS, any time you know enough sides to find an angle directly. The law of sines (a/sin A = b/sin B = c/sin C) handles ASA and SSA, any time angles and their opposite sides are in play. This tool picks the right one automatically based on what you've entered.
Triangle solver FAQ
What do SSS, SAS, ASA and SSA mean?
They describe which three measurements you already know. SSS is all three sides. SAS is two sides and the angle between them. ASA is two angles and the side between them. SSA is two sides and an angle that is not between them, the "ambiguous case", covered below.
Why does SSA sometimes give two different answers?
Knowing two sides and a non-included angle doesn't always pin down a unique triangle, geometrically, two different triangles can share those exact same three measurements, mirrored around the side you don't have information about. This tool checks for that case and shows both valid triangles when it happens, rather than silently picking one.
Why did my SSA input return no solution?
If the given angle and the two side lengths simply can't form a closed triangle (the side opposite your known angle is too short to ever reach the third vertex), no triangle exists at all. This is a real, common outcome for SSA, not a calculation error.
How is the area calculated once the triangle is solved?
Once all three sides are known, this uses Heron's formula: with s as half the perimeter, area = √(s(s−a)(s−b)(s−c)). It works for every solved triangle regardless of which combination of sides and angles you started from.
Is my data private?
Yes. Every calculation and the diagram run in your browser. Nothing you enter is sent to a server.
Free triangle solver by ANUPRESS
Two laws, four cases
The law of cosines extends Pythagoras to triangles without a right angle and handles anything where you know enough sides to solve directly: three sides (SSS), or two sides and the angle between them (SAS). The law of sines instead relates each side to the sine of its opposite angle, and handles the cases built around angles: two angles and the side between them (ASA), or two sides and an angle that isn’t between them (SSA). Picking the right law for what you actually know is most of the battle, this tool does that automatically based on which mode you select.
Why SSA can honestly have two right answers
SSA is called the “ambiguous case” for a real geometric reason, not a calculation quirk: knowing two sides and a non-included angle can describe two genuinely different triangles, mirrored around the side you don’t have a fixed angle for, both perfectly valid. Picture swinging the unknown side like a compass from a fixed point, it can cross the target length at two different points, giving two different closing angles. This tool checks for exactly that condition and shows both triangles rather than silently returning just one and hiding the ambiguity.
When SSA gives you nothing at all
The other SSA outcome is no triangle whatsoever: if the side opposite your known angle is too short to ever swing around and reach the third vertex, no closed shape is possible with those measurements, full stop. That’s not a rounding error or a bug, it’s the correct answer, the same way asking for a triangle with sides 1, 2, and 10 has no answer because the two short sides can never reach far enough to meet the long one.