What is the triangle inequality?

How to test three lengths

A helpful way to think about this is that the straight path between two points is the shortest route. Going from one endpoint to the other by detouring through the third vertex must be longer than the direct side.

This theorem matters both for checking whether a triangle can exist and for judging whether a calculated side length makes sense. It acts as an early realism test in triangle problems.

Side lengths 4, 5, and 8 pass because 4 + 5 is greater than 8. Lengths 2, 3, and 5 fail because the two shorter sides only reach equality, making a flat line instead of a genuine triangle. The inequality asks whether the shape can actually close.

• The sum of any two side lengths in a triangle must be greater than the third side.
• The inequality must hold for each side of the triangle, not just for one chosen side.
• If a side equals the sum of the other two, the figure collapses into a straight line and is no longer a triangle.
• The converse is practical: three lengths can form a triangle only if every pair sums to more than the remaining length.

Where triangle inequality helps

• Use the triangle inequality to test whether a proposed triangle is possible before drawing it.
• Use it to check whether a computed side length is reasonable in a geometry problem.
• Use it in proofs and reasoning about shortest paths and side comparisons.

Three triangle-inequality checks

• Do not check only one pair of sides; all three comparisons matter.
• Do not accept the equality case as a valid triangle.
• Do not forget that the theorem is about lengths, so unit consistency still matters.