Math Solver Free

Triangle Calculator

Solve SSS, SAS, or ASA — every side, angle, area, and perimeter.

Side a
3
Side b
4
Side c
5
∠A
36.87°
∠B
53.13°
∠C
90°
Area
6
Perimeter
12

Step 1: Angle A (opposite side a) via law of cosines: cos A = (b² + c² − a²) / 2bc = 36.87°

Step 2: Angle B similarly = 53.13°, then C = 180° − A − B = 90°

Step 3: Semi-perimeter s = (a + b + c) / 2 = 6; area = √[s(s−a)(s−b)(s−c)] = 6

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How to Solve a Triangle

“Solving” a triangle means finding every unknown side and angle from the ones you know. You always need three pieces of data, and at least one of them must be a side — three angles alone only fix the shape, not the size.

The Three Cases

  • SSS (three sides): angles come from the law of cosines, e.g. cos A = (b² + c² − a²) / 2bc.
  • SAS (two sides + included angle): the third side comes from the law of cosines, then the law of sines finishes the angles.
  • ASA (two angles + included side): the third angle is 180° minus the other two, then the law of sines gives the remaining sides.

Watch Out For

Not every trio of numbers makes a triangle: sides 2, 3, 10 violate the triangle inequality. And SSA (two sides and a non-included angle) is the famous ambiguous case that can yield zero, one, or two valid triangles — a deeper topic for another day.

Solving the equations behind these laws by hand? Try our equation solver or the scientific calculator.

Frequently Asked Questions

What information do I need to solve a triangle?

Any three pieces of information with at least one side: three sides (SSS), two sides and the included angle (SAS), or two angles and the included side (ASA). Choose the matching tab above and enter your values.

How is the area of a triangle calculated here?

Once all three sides are known, the calculator uses Heron's formula: area = √[s(s−a)(s−b)(s−c)] where s is the semi-perimeter. For SAS input it can also use ½ab·sin(C).

What if my sides cannot form a triangle?

The calculator checks the triangle inequality (the sum of any two sides must exceed the third). If your values fail it — for example 2, 3, 10 — you will see a clear error instead of nonsense results.

Is this triangle calculator free?

Yes — every tool on Math Solver Free is free with no sign-up, and all calculations run in your browser.