Pythagorean Theorem Calculator
The theorem machine — hypotenuse from legs, a missing leg from the hypotenuse, or the converse verdict: is it right at all? Triples named, irrationals kept honest.
Pythagorean Theorem Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Right triangles only — the general SSS question lives on the triangle page
- Two-dimensional; 3D distances chain the theorem twice
In short: Legs 3 and 4: 3² + 4² = 9 + 16 = 25 and √25 = 5 — an exact triple, no rounding anywhere. The same lever lifts the other way: hypotenuse 5 with leg 4 forces the missing leg to be √(25 − 16) = 3. The converse is its own question: 6-8-10 passes (100 = 100, right), while 3-4-6 fails with the numbers shown (9 + 16 = 25, but 6² = 36 — the corner has swung open past 90°). And legs 1, 1 give √2 ≈ 1.414214: not every right triangle is a triple, and the page says so rather than rounding a fake whole number.
Formula
a² + b² = c²
c = √(a² + b²) · leg = √(c² − other²)
the theorem runs both directions: build the hypotenuse from the legs, recover a leg from the rest, or test whether any three sides form a right triangle at all.
Worked Example
- Square. each side squared — the theorem lives in the squares, and the page prints every one.
- Combine. legs add; recovering a leg subtracts. A leg mode refuses c ≤ leg, because the hypotenuse must exceed each leg for the subtraction to mean anything.
- Verdict. converse mode compares c² against a² + b² with both numbers shown: equal is RIGHT, larger is OBTUSE, smaller is ACUTE.
3-4: 9 + 16 = 25 → c = 5, an exact triple. 1-1: √2 ≈ 1.414214 — honest irrationality. Converse: 6-8-10 right (100 = 100); 3-4-6 obtuse (25 against 36, the corner past 90°). Families: 5-12-13 and 8-15-17 verified live; scale any triple by k and it stays a triple — 15-20-25 is 3-4-5 × 5.
Strengths & Limits Of This Model
Where this engine is strong
- All three questions — build, recover, verify — on one grammar
- The converse names both square sums, not just a verdict
Where it stops
- No angle outputs (the trig page solves those)
- No triple SEARCH — only naming the one you hit
Practical Use Cases
Squaring up
the 3-4-5 check that true corners on site
Shortest paths
diagonal walks across grids and fields
Homework
squares printed, verdicts compared, triples named
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Pythagorean Theorem Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What exactly does the theorem say?
In a RIGHT triangle, the square on the hypotenuse equals the sum of the squares on the two legs. It is a statement about AREAS of squares built on the sides, which is why the page lives in the squares: 9, 16 and 25 are the whole story, and the 5 is their square root.
How is the converse different from the theorem?
The theorem assumes a right angle and predicts the sides; the converse assumes three sides and asks whether the right angle exists. Both directions are true, and the converse is the practical one: it is how a tape measure squares a corner — 6-8-10 across a diagonally measured frame proves it rectangular.
Why does a leg need c to be strictly bigger?
The hypotenuse is the longest side by definition — it spans the opened corner. If c equals a leg the triangle degenerates (the subtraction gives zero); if c is smaller the subtraction goes negative, and a negative under the root is the geometry saying no such triangle exists. The page refuses instead of printing the wreckage.
What is a Pythagorean triple?
Three whole numbers that satisfy the theorem exactly — 3-4-5, 5-12-13, 8-15-17 and their multiples. Whole legs landing on a whole hypotenuse is rare enough to be worth naming: the page checks your result and calls the triple when it sees one. Scaling preserves triples: 15-20-25 is just 3-4-5 grown by five.
Why print √2 instead of rounding to 1.41?
Because 1.41 is a LIE about the number: √2 is irrational, provably never a fraction, and the legs 1-1 were the classic proof that not every right triangle is a triple. The page prints 1.414214 for the display and says the exact value is √2 — honesty about the irrational is part of the method.
Does the theorem work in 3D?
Yes, twice. The straight-line distance between two 3D points applies Pythagoras to get the horizontal shadow, then again to lift it by the height — which is exactly how the distance formula page extends to three squares under one root.
What does an obtuse or acute verdict mean physically?
If c² beats a² + b² the corner opposite c has swung OPEN past 90° (obtuse); if it falls short, the corner is pinched (acute). For 3-4-6 the squares read 25 against 36 — the 6 side is too long for a right angle, and the page shows both numbers so the verdict carries its evidence.
Can I use decimals?
Freely — legs 1.5 and 2 give an exact 2.5 (the 3-4-5 triple scaled by half). Triples are a whole-number curiosity, not a requirement; the theorem holds for every positive real pair.