How this instrument works
Right triangles hide a piece of exact arithmetic: square the two legs, add the squares, and the sum is precisely the square of the hypotenuse — a² + b² = c². The relation holds for every right triangle, from a drafting square to a surveying baseline, and it is exact, not approximate. That makes it the standard way to recover a third side from any two, and the quiet engine behind distance formulas in coordinate geometry, vector magnitudes, and every diagonal you have ever measured across a rectangle.
The theorem runs equally well in reverse. If you know the hypotenuse and one leg, subtract the squares instead of adding them: b = √(c² − a²). One caution comes built in — the hypotenuse is always the longest side, so c must exceed the known leg or the square root has nothing real to return. This sheet checks that for you and flags an impossible pair before it computes.
Nothing in the formula cares about units. Measure in millimetres, feet, or grid squares; as long as both known sides share a unit, the answer arrives in it. Scale a right triangle up or down and the relation survives untouched, which is why one worked 3-4-5 rope trick has set square corners for four thousand years, from Egyptian field boundaries to modern deck framing.
- Pick the mode: Solve c when both legs are known, Solve b when you have one leg and the hypotenuse.
- Enter Leg a, then either Leg b or Hypotenuse c. Use one consistent unit for both — the result comes back in the same one.
- In Solve b mode, keep Hypotenuse c longer than Leg a; the sheet flags any pair where the hypotenuse is not the longest side.
- Read the third side from the result field, shown to four decimal places, with the inputs recorded in the log line.
Worked example — the 3-4-5 rope triangle
Take the classic pair: Leg a = 3 and Leg b = 4. Square each — 9 and 16 — add to get 25, and take the root: c = √25 = 5, exactly. This is the 3-4-5 triple, the smallest right triangle with three whole-number sides, and the reason a knotted rope with sides of 3, 4, and 5 units pulls itself into a perfect square corner.
Now run it backwards. Switch the dial to Solve b, enter Leg a = 3 and Hypotenuse c = 5: b = √(5² − 3²) = √(25 − 9) = √16 = 4. The next triple up is 5-12-13 — feed in legs 5 and 12 and the instrument returns 13, another exact integer with no rounding involved.
Questions
What is the Pythagorean theorem formula?
a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse. To find the hypotenuse, take the square root of the sum of the squared legs: c = √(a² + b²). To find a missing leg, subtract instead: b = √(c² − a²). Both forms are exact rearrangements of the same identity.
Does the Pythagorean theorem work on every triangle?
No — only on right triangles, ones containing a 90° angle. For any other triangle the relation picks up a correction term and becomes the law of cosines: c² = a² + b² − 2ab·cos(C). When the angle C is exactly 90°, the cosine term vanishes and the law of cosines collapses back into a² + b² = c².
Which side is the hypotenuse?
The side opposite the right angle — always the longest of the three. The other two sides, the legs, are the ones that meet to form the 90° corner. In this instrument the legs are a and b; the hypotenuse is c. If the value you enter for c is not the longest side, the sheet flags it rather than returning an imaginary number.
Why does the calculator reject a hypotenuse shorter than the leg?
Because b = √(c² − a²) asks for the square root of c² − a², and if c is not longer than a that quantity is zero or negative — no real triangle has those measurements. Geometrically the hypotenuse must exceed either leg on its own (though never their sum), so a shorter c means the two numbers cannot belong to the same right triangle.
What is a Pythagorean triple?
Three whole numbers that satisfy a² + b² = c² exactly. The smallest is 3-4-5, followed by 5-12-13, 8-15-17, and 7-24-25; multiply any triple by a constant and it still works, so 6-8-10 and 9-12-15 count too. Triples are handy for checking square corners with nothing but a tape measure — no square root required.
Do the sides need to be in specific units?
No. The theorem is a pure statement about lengths, so any unit works — metres, inches, pixels, grid squares — provided every side uses the same one. Enter the two known sides in centimetres and the third side comes back in centimetres. Mixing units is the one way to get a wrong answer out of a correct formula.