Three sides (SSS)
A 5-6-7 triangle has angles of about 44.4°, 57.1° and 78.5°, an area of about 14.70, and is scalene.
Triangle solver
Result
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How it works
The law of cosines generalizes the Pythagorean theorem to any triangle, not just right ones, and is what recovers a missing side or angle when three sides (SSS) or two sides and the included angle (SAS) are known. The law of sines instead relates every side to the sine of its opposite angle, which is the natural tool once two angles and a side (ASA) are known.
Once all three sides and angles are known, area follows directly from two sides and the angle between them, and the triangle is classified by its sides (equilateral, isosceles, scalene) and by its largest angle (acute, right, obtuse).
Formulas used
Law of cosines
c² = a² + b² − 2ab·cos(C)
Law of sines
a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C)
Area from two sides and the included angle
Area = ½ · a · b · sin(C)
Worked examples
Two sides and the included angle (SAS)
Sides of 5 and 6 with a 60° angle between them give a third side of about 5.57 and an area of exactly 12.99.
Assumptions and limits
- The triangle lies in a flat (Euclidean) plane.
- Inputs form a valid triangle — impossible combinations are flagged as errors rather than solved.
Frequently asked questions
Why do I need to pick SSS, SAS or ASA?
Because the solving method depends on which pieces of information you already have — three sides need the law of cosines throughout, while two angles and a side is most directly solved with the law of sines.
What if the three sides I enter cannot form a triangle?
The triangle inequality requires any two sides to add up to more than the third — if that fails, no triangle exists with those measurements, and the calculator flags it rather than showing an impossible result.
How is the triangle classified?
By its sides — equilateral (all equal), isosceles (two equal) or scalene (none equal) — and separately by its largest angle: acute if under 90°, right if exactly 90°, obtuse if over 90°.
Does this handle the ambiguous case (SSA)?
Not in this version — two sides and a non-included angle can sometimes match two different triangles, which needs extra handling this tool does not cover; use SSS, SAS or ASA instead, which always have a single solution.
Updated
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