Triangle Calculator

The triangle inequality decides whether a triangle exists at all

Any two sides added together must exceed the third. Sides of 3, 4 and 9 cannot close into a triangle no matter how you arrange them — the two short sides fall short of reaching across the long one. The calculator checks this before anything else, because every other formula happily returns a plausible-looking number for an impossible shape.

Which formula gets used, and why

SSS uses Heron's formula for the area — √(s(s−a)(s−b)(s−c)) where s is half the perimeter — and the law of cosines for the angles. It never needs a height, which is what makes it so useful for surveyed or measured triangles.

SAS uses area = ½ab·sin(C), then the law of cosines to find the third side. ASA finds the third angle by subtraction (the three must total 180°) and then the law of sines for the remaining sides.

Right triangle mode is SAS with C fixed at 90°, which collapses the law of cosines to a² + b² = c² — Pythagoras is not a separate theorem so much as the right-angled corner of a more general one.

Classifying by angle and by side

Compare the square of the longest side against the sum of the squares of the other two. Larger means obtuse, equal means right-angled, smaller means acute. By side lengths a triangle is equilateral (all three equal), isosceles (two equal) or scalene (none equal). Both classifications are reported, because a triangle can be, for example, both isosceles and obtuse.

Circumradius and inradius

R = abc / 4×Area is the radius of the circle through all three corners; r = Area / s is the radius of the circle that fits snugly inside touching all three sides. They matter far beyond geometry homework — in structural layout, in mesh generation, and in the standard test for how badly shaped a triangle is.

Frequently Asked Questions