How this instrument works
This calculator checks a single claim: do three known side lengths, a, b, and c, satisfy the Pythagorean relation exactly? It computes test = a² + b² − c², where c is entered as the longest of the three. A result of exactly zero confirms a genuine right triangle; anything else means the three lengths don't meet at a square corner, even if they still form a perfectly valid triangle.
A sibling tool on this site runs a related check but hunts for the longest side itself, using a max() term so the three lengths can be entered in any order at all. This page skips that search — it trusts that you've already sorted the numbers and placed the largest one into Side c, then runs the plain subtraction a² + b² − c² directly. That makes it the quicker route specifically when a triple is already suspected, like 3-4-5 or 5-12-13, and you just want the confirmation.
A nonzero test value is not a failure of the triangle, only of the right angle. Take 5, 6, and 7: those three lengths close into a completely ordinary triangle — every side shorter than the sum of the other two — but 25 + 36 − 49 comes out to 12, not zero, so no corner inside it is a right angle. The test isolates that one geometric fact and nothing more.
- Enter Side a, Side b, and Side c — place the longest of your three measurements into Side c.
- Read Test value: exactly 0 means the three sides form a right triangle.
- Any nonzero Test value means no right angle, even if the three lengths still make a valid triangle.
- Try known triples such as 3, 4, 5 or 5, 12, 13 and watch Test value land on exactly 0 both times.
Worked example — testing 3-4-5 and 5-6-7
Enter a = 3, b = 4, c = 5: test = 3² + 4² − 5² = 9 + 16 − 25 = 0 exactly — a right triangle, and the smallest whole-number triple that pulls this off, the same shape rope-stretchers used to square corners long before algebra existed. Try a = 5, b = 12, c = 13 next: test = 25 + 144 − 169 = 0 again, a second classic triple confirmed just as cleanly.
Now try a = 5, b = 6, c = 7: test = 25 + 36 − 49 = 12, a firm nonzero result. Those three lengths still close into a real triangle — 5 + 6 comfortably exceeds 7 — but no corner inside it measures 90°, and the test value of 12 is the sheet's way of saying so without needing a protractor.
Questions
What does a Test value of exactly 0 mean?
It means the three side lengths you entered form a genuine right triangle: a² + b² equals c² exactly. With a = 3, b = 4, c = 5, test = 9 + 16 − 25 = 0, confirming the classic 3-4-5 triple.
Why does Side c specifically have to be the longest?
Because the formula a² + b² − c² only equals zero for a right triangle when c is the side opposite the right angle — always the longest of the three. Put a shorter side into the c field by mistake and the test will read a nonzero value even for a genuine 3-4-5 triangle, since the arithmetic is no longer checking the right relationship.
Is 5, 6, 7 still a real triangle even though the test isn't zero?
Yes. Any three lengths where each one is shorter than the sum of the other two close into a valid triangle, and 5, 6, and 7 pass that check easily. The nonzero test value of 12 only says this particular triangle has no right angle in it — it says nothing about whether the shape itself is valid.
How is this different from a checker that finds the longest side automatically?
A related tool on this site accepts three lengths in any order and searches for the largest one itself before running its own version of the test. This calculator skips that search: it assumes you already know which side is longest, expects it in the Side c field, and runs a single subtraction, a² + b² − c², directly — faster when a specific triple is already the one you're checking.
What is a Pythagorean triple?
Three positive whole numbers, a, b, and c, that satisfy a² + b² = c² exactly, with no rounding involved anywhere. 3-4-5 is the smallest example, 5-12-13 the next well-known one, and both return a Test value of exactly 0 on this page.
Can a genuine right triangle ever show a Test value that isn't quite 0?
Only with decimal side lengths rather than a clean integer triple, and only by a tiny floating-point residue — something like 0.0000000003 instead of a flat zero. Whole-number triples like 3-4-5 or 5-12-13 always land on exactly 0, with nothing left to round away.