Triangle Solver
Solve a triangle from any three parts, showing both answers where two sides and a non-included angle admit two valid triangles.
Fill in what you know — any three values, at least one of them a side. Leave the rest blank.
Each angle sits opposite the side of the same letter.
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Why there is sometimes more than one
- Two sides and the angle BETWEEN them give exactly one triangle. Two sides and an angle that is not between them may give none, one, or two — and when there are two they are both real, both satisfying every law. Reporting one of them without saying so is not a rounding difference; it is half the answer.
- It is not a rare corner either. Of 40,000 random questions of that shape — both sides between 1 and 20, the angle between 1 and 89 degrees — 12,072 had no triangle at all, 19,981 had one, and 7,947 had two. That is 19.9 per cent, and every one of them is a question a calculator showing a single answer gets half right. Those counts are computed when this page is built, not quoted, because the proportion depends entirely on where the numbers come from.
- That happens because an angle and its supplement have the same sine. Working an angle back from its sine cannot tell 40 degrees from 140, so both have to be tried, and whichever still leaves room for a third angle survives.
- When nothing fits, the tool says which rule the numbers break — a side too short to reach, two sides that cannot reach the third, or two angles that leave nothing for a third. It never returns a triangle whose angles fail to add to 180.
- Every triangle reported is checked against the laws it should obey before you see it: the angles add to 180, the sides close up, the law of sines gives one ratio for all three pairs, the law of cosines holds on all three, and two different area formulas agree. Those are exact identities, so they either hold or the answer is wrong.
- A right triangle needs only two parts, but an angle on its own is not enough — it fixes the shape and says nothing about the size. Two angles have the same problem, which is why every case here needs at least one length.
How to use
- Enter any three known sides or angles.
- Read the remaining parts, plus area and perimeter.
- Check for a second solution in the ambiguous case.
- Confirm the parts you entered can form a triangle at all.
Frequently asked questions
What is the ambiguous case?
When you know two sides and an angle not between them, there can be two different triangles that fit, one with an acute angle and one with an obtuse one. Both are genuine solutions, and a solver reporting only one is quietly hiding half the answer — which is why this one shows both.
When do I use the law of sines rather than the law of cosines?
The law of sines when you have an angle paired with its opposite side; the law of cosines when you know all three sides, or two sides and the angle between them. The law of cosines is also more numerically reliable, and generalises Pythagoras to non-right triangles.
Which combinations of three parts determine a triangle?
Three sides, two angles and any side, or two sides and the included angle each give exactly one triangle. Two sides and a non-included angle may give none, one or two. Three angles give the shape but not the size, since any scaled copy has the same angles.
What is the triangle inequality?
Any two sides must sum to more than the third. If they do not, no triangle exists — the two shorter sides cannot reach across. It is the first thing to check when a solver reports no solution for measurements you are confident about.
How is the area found from three sides?
By Heron's formula, using the semi-perimeter. It is elegant but numerically poor for very thin triangles, where it subtracts nearly equal quantities and loses precision. A rearranged version that sorts the sides first is stable and is what careful implementations use.
Why do my angles not sum to exactly 180?
Rounding in the display. The underlying calculation preserves the sum, but each angle shown to a fixed number of decimal places is rounded independently, so the printed values can be a hundredth out. It is a display artefact, not an error in the solution.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.