Give a triangle its three sides and everything else is determined — every angle, the area, the shape's very possibility. This calculator solves the SSS case completely, plus the right-triangle shortcut where two legs are all you know.
First, can the triangle exist?
The triangle inequality is the bouncer at the door: any two sides together must exceed the third. Sides 2, 3, 7 never close — 2 + 3 falls short of spanning 7 — and the calculator refuses politely instead of producing NaN angles. (Degenerate "triangles" where the sum exactly equals the third side are flat lines, also rejected.)
Angles from the law of cosines
Each angle is recovered from the side facing it: cos(C) = (a² + b² − c²) / 2ab. For the classic 3-4-5: the angle opposite the 5 works out to exactly 90° — the calculator labels it a right scalene triangle, and Pythagoras nods. The three angles always land on 180°, a built-in sanity check.
Area without heights: Heron's formula
No altitude needed — the 2,000-year-old formula runs on sides alone. For 3-4-5: s = 6, area = √(6·3·2·1) = 6. It generalizes to any valid triangle, and its failure mode (a negative under the root) is precisely the inequality check above.
The right-triangle mode
Two legs in, hypotenuse out via a² + b² = c² — plus the same full solution. Carpenters know the trick as 3-4-5 corner-squaring; the 45-45-90 (hypotenuse = leg × √2) and 30-60-90 (1 : √3 : 2) families cover most of trigonometry class. The scaled sketch redraws live, so an obtuse triangle looks obtuse. Coordinates instead of lengths? The slope calculator measures the distance between two points, and the square root calculator simplifies the radicals these formulas produce.
Angles in degrees; validity checked before solving.