Triangle Solver (SSS)
Solve any triangle from its three side lengths (SSS). Enter sides a, b and c to get all three interior angles via the law of cosines, plus the area from Heron's formula and the perimeter.
Enter Values
How to use this calculator
- Enter the three side lengths a, b and c in the same unit (all cm, all m, etc.).
- Make sure the sides can form a triangle — each side shorter than the sum of the other two.
- Read angles A, B and C (each opposite its same-letter side), the area and the perimeter.
How it works
With three sides known, each angle comes from the law of cosines: angle A = acos((b² + c² − a²) ÷ (2bc)), and similarly for B and C. The three angles always add to 180°, which is a handy check. The area uses Heron's formula: with semi-perimeter s = (a + b + c) ÷ 2, area = √(s(s−a)(s−b)(s−c)). The perimeter is simply a + b + c. Area is in squared length units and the perimeter in the same length units as the sides.
Worked example
Worked example. For a 3-4-5 triangle: angle A = acos((16 + 25 − 9) ÷ 40) = acos(0.8) = 36.87°, angle B = acos((9 + 25 − 16) ÷ 30) = acos(0.6) = 53.13°, and angle C = acos((9 + 16 − 25) ÷ 24) = acos(0) = 90° — a right angle, as expected. The semi-perimeter is 6, so the area is √(6 × 3 × 2 × 1) = 6 square units and the perimeter is 12 units.
Common mistakes
- Entering sides that break the triangle inequality (e.g. 1, 1, 5) — no triangle exists, so the calculator returns an error.
- Mixing units between the three sides; keep a, b and c all in the same unit.
- Reading an angle against the wrong side — each angle is opposite the side with the same letter (A opposite a).
Frequently asked questions
Why does the calculator need all three sides?
This is the SSS (side-side-side) case. Three sides fix a unique triangle, so the law of cosines can find every angle and Heron's formula gives the area without needing any angle input.
What units do the area and perimeter use?
They inherit the length unit of the sides you enter. If the sides are in metres, the perimeter is in metres and the area is in square metres.
Related tools
- Polygon Interior Angle Calculator
- Coordinate Geometry Calculator
- Distance Formula Calculator
- Inverse Trig (arcsin, arccos, arctan) Calculator
- Angle Sum Calculator
- Circle Equation Calculator
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Tip: Enter any known values to calculate the remaining results.
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