Triangle Calculator

Calculate triangle area, perimeter, and angles free online from three side lengths. Real Heron's formula and law of cosines, with triangle validity checking.

Side a

Side b

Side c

Browser only -- real Heron's formula and law of cosines.

How to Use This Tool

Enter all three side lengthsAny valid triangle

See area, perimeter, and anglesCalculated via real geometric formulas

Get a validity checkTriangle inequality verified automatically

Triangle Calculator is a free online tool that works entirely in your browser — no signup, no data uploaded, no limits.

Heron's Formula -- Area Without Needing Height

Most area formulas need a base and height, but Heron's formula calculates a triangle's area purely from its three side lengths, using the semi-perimeter as an elegant intermediate step. This calculator applies that exact formula rather than requiring you to separately determine height first.

The Law of Cosines for Angles

Generalizing the Pythagorean theorem beyond right triangles, the law of cosines relates all three sides to any one angle, letting this calculator solve for all three interior angles directly from the side lengths alone, without needing any angle measurement as a starting input.

Validity Checking Prevents Nonsensical Results

Not every three numbers can form an actual triangle -- the triangle inequality (any two sides must sum to more than the third) determines this. This calculator checks that condition first, flagging invalid combinations rather than attempting to calculate geometry for an impossible shape.

Frequently Asked Questions

Heron's formula: Area = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (half the total perimeter) -- a genuinely elegant method that finds area without needing to know any angle or height directly.

The law of cosines, rearranged to solve for each angle: cos(A) = (b²+c²-a²)/(2bc), applied once per angle using the appropriate side lengths -- a generalization of the Pythagorean theorem that works for any triangle, not just right triangles.

The triangle inequality theorem: the sum of any two sides must exceed the third side. If this fails for any combination, the three lengths cannot form a real triangle, and the calculator flags this rather than producing a nonsensical result.

If two sides summed were shorter than the third, they couldn't physically stretch far enough to connect to the third side's endpoints -- imagining trying to draw it makes the impossibility visually obvious, which is exactly what the inequality formalizes.

This works for any valid triangle, including right triangles as a special case where one angle happens to calculate to exactly 90 degrees -- no special-casing needed since the general formulas handle it correctly.