Formula and Method for the Triangle Inequality Theorem
The Triangle Inequality Theorem says that for any triangle with side lengths a, b, and c, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side: a + b > c, a + c > b, and b + c > a. If even one of these three conditions fails, the given lengths cannot be joined into a closed triangle — the two shorter sides simply can't reach across to meet the third.
How the calculation works
Enter your three side lengths. Because a triangle's longest side is the hardest one to "reach" with the other two, it is enough to compare the two shortest sides' sum against the longest side: if longest < sum of the other two, all three inequalities are automatically satisfied and the triangle is valid. This calculator runs that check plus the full three-way comparison, then reports the valid range for a missing third side as |a − b| < c < a + b given any two known sides. When the triangle is valid, it also classifies the shape by side length (equilateral, isosceles, or scalene) and by angle (acute, right, or obtuse) using the Law of Cosines relationship: comparing c² to a² + b² for the longest side c tells you whether the angle opposite it is less than, equal to, or greater than 90°.
Common mistakes
- Treating equality as valid: if a + b = c exactly, the three lengths are collinear (a straight line with zero area), not a true triangle — the inequality must be strict (>), not (≥).
- Checking the wrong pair: the inequality that matters most is always the two shortest sides versus the longest; checking a random pair can miss the failure.
- Confusing this with the Pythagorean theorem: a² + b² = c² only applies to right triangles and tests for a specific angle, while the triangle inequality tests whether a triangle can exist at all, for any angle.
- Mixing units: all three side lengths must be measured in the same unit before comparing them, or the check is meaningless.
Real-world applications
- Construction and carpentry use it to confirm that three measured cuts (rafters, braces, trusses) can actually be joined into a rigid triangular frame.
- Structural and mechanical engineering rely on it when designing trusses and linkages, since an invalid triangle means the members cannot physically connect.
- Computer graphics and mesh-generation software validate triangle geometry with this test before rendering or simulating a 3D surface.
- GPS trilateration and surveying use the inequality to sanity-check that three measured distances are geometrically consistent before trusting a computed position.