How this instrument works
The Triangle Inequality Theorem says that any two sides of a triangle, added together, must be longer than the third — for all three pairings at once. This sheet collapses that three-part check into a single figure: test = min(a+b−c, a+c−b, b+c−a). Only the smallest of the three differences can ever fail, because it always belongs to the pairing that leaves out the longest side, so one number is enough to settle the question.
The geometric reason is straightforward: picture two rods of length a and b hinged at one end. However you swing them, the free ends can never end up farther apart than a+b, since a straight line is the shortest path between two points. If the third rod, c, is longer than that maximum reach, the hinge can never swing far enough to meet it, and no triangle closes. When c equals a+b exactly, the hinge straightens flat — a degenerate triangle with zero area, not a real one.
That boundary case is worth sitting with: sides 2, 3, and 5 push the test to precisely 0, because 2+3=5 on the nose. It looks like it should count, but a flat line segment has no interior angle to measure and no area to compute, which is why the theorem is stated with a strict greater-than. Nudge any one of those three lengths by even a tiny amount and the result tips to positive or negative — the boundary has no width.
- Type your first measured length into Side a.
- Add the second length in Side b — the order you fill the three fields in makes no difference to the result.
- Finish with the third length in Side c.
- Read Test value (positive = valid triangle): a positive number means those three lengths form a genuine triangle, zero means they only just meet in a flat line, and a negative number means they cannot meet at all.
Worked example — carpentry stock at 3, 4, and 5 units
Suppose three cut lengths come off the rack at a = 3, b = 4, c = 5 units. The sheet checks all three pairings — 3+4−5 = 2, 3+5−4 = 4, 4+5−3 = 6 — and keeps the smallest. Test value reads 2.0, a clean positive, so these three offcuts will hinge together into a genuine closed triangle. This particular triple has a second claim to fame: it also satisfies 3²+4²=5², so the shape it forms happens to carry a right angle opposite the length-5 piece.
The 2.0 is not merely a pass mark, it's the margin. Piece c could grow by up to two units, or pieces a and b could shrink by a combined two units, before the shape flattens to nothing. Contrast that with 1, 1, and 5 fed into the same three fields: 1+1−5 comes out to −3, an unambiguous fail, since the two short pieces fall three units short of ever reaching across to the long one.
Questions
What does a negative test value mean?
It means the three lengths cannot form a triangle. The two shorter sides, even stretched into a straight line, still fall short of reaching both ends of the longest side, so no vertex can close the shape. For sides 1, 1, and 5 the test comes out to −3 — a three-unit gap that no amount of adjusting angles can bridge.
What's special about landing on precisely zero?
Zero marks the exact boundary between a real triangle and no triangle at all. It means the two shorter sides sum to precisely the length of the third, so the figure has flattened into a straight line with no area. Sides 2, 3, and 5 land exactly there — 2+3=5 — which is why the theorem is written with a strict greater-than rather than greater-than-or-equal.
Do all three pairings of sides really need checking?
Only the tightest one can ever fail, and the min() in the formula finds it automatically. Whichever side is longest is the only side at risk of being outreached, so once the sum of the other two beats that longest side, the remaining two pairings are already guaranteed to pass. This sheet still evaluates all three and reports the smallest, so nothing depends on you knowing which side is longest beforehand.
How does this relate to the Pythagorean theorem?
The two checks sit at different stages of the same problem. This one asks whether a triangle can exist at all, before anything about its angles is decided; the Pythagorean theorem only applies afterward, to test whether an already-valid triangle happens to carry a right angle. The 3-4-5 triple passes both — a positive test of 2 here, and 3²+4²=5² separately — but most valid triangles never satisfy a²+b²=c² and still pass this test easily.
Where does this check actually get used?
Anywhere three measured or specified distances get assembled into a shape: cutting stock to a cut list, validating triangulated survey or GPS distances, or checking that fabricated parts will actually mate at three fixed points. If the three numbers came from independent measurements, a negative or near-zero test value is often the first sign that one of them was recorded wrong.
Is the triangle inequality only about triangles?
No — the same principle, that a direct path is never longer than a detour through a third point, appears throughout mathematics under the same name, governing ordinary distances, vector norms, and abstract metric spaces alike. The three-sides-of-a-triangle version checked here is the original geometric case the whole family of rules is named after.