Triangle Inequality Theorem Calculator

The fundamental premise of the triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must always be strictly greater than the length of the remaining third side. Mathematically, for a triangle with sides labeled a, b, and c, three separate inequalities must all hold true simultaneously: a + b > c, a + c > b, and b + c > a. If even one of these three conditions fails, the segments are too short or disjointed to close the shape at the vertices, meaning no valid triangle can exist.

Imagine you have three wooden dowels measuring 7 inches, 10 inches, and 15 inches, and you want to know if they can form a stable triangular frame. To test this using the theorem, we check all three potential combinations. First, check the two shorter sides added together against the longest side: 7 + 10 = 17, which is greater than 15. Second, check 7 + 15 = 22, which is greater than 10. Third, check 10 + 15 = 25, which is greater than 7. Because all three inequalities evaluate as true, these specific lengths successfully form a valid triangle.

When testing side lengths, always remember to verify the combination involving the two shortest sides, as that is the test most likely to fail in non-working triangles. Additionally, keep in mind that a sum equal to the third side (such as 3 + 4 = 7) results in a degenerate triangle—a flat line segment with zero area—rather than a true multi-dimensional polygon. Always ensure your measurements are converted to the same unit before running the calculation.