Triangle Solver: SSS, SAS and ASA with Area

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How to use this tool

Tell it what you know: three sides, two sides with the angle between them, or two angles with the side between them. It returns all six elements, the area and the perimeter. Anything geometrically impossible is named and rejected, whether that is a triangle inequality violation or two angles that already sum past 180.

The labels follow the standard convention: angle A sits opposite side a, B opposite b, C opposite c. In the SAS mode, the angle you enter is the included one, between the two sides you enter. That is the detail that makes SAS unambiguous, because two sides with a non included angle can describe two different triangles.

What the solver actually does

SSS runs the law of cosines twice to find two angles, and takes the third from the 180 degree sum. SAS runs the law of cosines forward to get the missing side, then falls back to the SSS path. ASA finds the third angle first, then scales the two missing sides with the law of sines.

The area always comes from Heron on the final three sides, never from the inputs directly. That is a deliberate cross check: if the solving went wrong, Heron on inconsistent sides produces a visibly wrong or impossible area, so agreement between the two is quiet evidence the solution hangs together.

A worked check you can repeat on paper

Sides 3, 4 and 5. The law of cosines for the angle opposite the 5: cos C is 9 plus 16 minus 25 over 24, which is zero, so C is exactly 90 degrees. The other two follow as 36.87 and 53.13, summing to 180. Area by Heron: semi perimeter 6, root of 6 times 3 times 2 times 1, exactly 6, which matches half of 3 times 4 as a right triangle must.

For ASA, take 60 degrees, a 10 unit side, 60 degrees. The third angle is 60, the triangle is equilateral, and both missing sides come back as exactly 10. Degenerate inputs make good tests precisely because the answer is known without trusting the tool.

Mistakes that catch people out

Using a non included angle as if it were included. Two sides and the angle not between them is the ambiguous SSA case, which can have two solutions. Fix: this tool takes the included angle; if yours is not, solve for it first.

Pairing a side with the wrong angle in the law of sines. Each side pairs with its opposite angle, never its adjacent one. Fix: follow the a opposite A convention strictly.

Degrees into a radian function. Every spreadsheet trig function expects radians. Fix: this tool converts internally, but hand checks need the factor pi over 180.

Trusting field measurements that fail the inequality. Taped sides of 5, 6 and 12 do not close, and the tape or the booking is wrong. Fix: treat the rejection as a data problem, not a tool problem.

How this was checked

The 3-4-5 returns its right angle to within floating point, the equilateral returns 60 everywhere, and the SAS case of 3 and 4 about 90 degrees returns a hypotenuse of exactly 5. Round trips were run, solving SSS and feeding the result back through SAS to confirm the same triangle emerges. Both rejection paths were tested: the inequality on 1, 2, 10 and the angle overflow on 120 plus 80.

Related Calculators and Reference

If you only need the area from three sides, the Area of 2D Shapes Calculator is quicker. Right triangles specifically are the business of the Distance and Pythagoras Calculator. Angle units, including gradians on European instruments, convert in the Angle Converter, and slope angles connect to the Grade to Degrees Converter. Also see the Trig Calculator: the sine, cosine and tangent behind each solved angle.