Triangle Calculator

Solve a triangle from its three side lengths — all three angles, area, and perimeter

Solving a triangle from its three side lengths alone — finding all three angles, area, and perimeter — relies on the Law of Cosines rather than the simpler right-triangle-specific Pythagorean relationship, since a general triangle (with no guaranteed right angle) needs a more broadly applicable trigonometric relationship to solve for its angles.

Why three sides alone are enough to solve the whole triangle

Given all three side lengths of a triangle, the triangle's shape is fully and uniquely determined — this is a basic geometric fact (side-side-side congruence), which means every other property (all three angles, area, perimeter) can be calculated directly from just those three lengths, without needing any angle measurement as a separate input. The Law of Cosines generalizes the Pythagorean relationship to work for any triangle, not just right triangles — solving it for each angle in turn using the three known side lengths recovers all three angles, after which area can be calculated using several equivalent formulas (Heron's formula, which uses only the three sides directly, is one common approach that doesn't require first finding the angles).

A worked example

Three side lengths that happen to satisfy the Pythagorean relationship (a² + b² = c²) describe a right triangle as a special case, while three side lengths that don't satisfy that relationship describe some other, non-right triangle shape — the Law of Cosines correctly handles both cases through the same general formula, reducing to the simpler Pythagorean case automatically when one angle genuinely is 90 degrees.

How this connects to your other geometry calculations

For triangles specifically known to be right triangles, the Right Triangle Calculator offers a more direct path using the simpler Pythagorean relationship plus basic trigonometry — use that dedicated calculator when a right angle is already known to exist, and this general triangle calculator when the triangle type isn't already established.

Common mistakes

Assuming three arbitrary side lengths automatically form a valid triangle, without checking the triangle inequality (the sum of any two sides must exceed the third), is a common oversight — not every combination of three positive lengths can actually form a closed triangle, and attempting to solve for angles using three lengths that violate this basic geometric constraint produces a meaningless or undefined result.