How this instrument works
Two of a right triangle's three sides are on the table here — the hypotenuse and one leg — with the third to be recovered. Hypotenuse and Known leg go in; Missing leg comes back as leg₂ = √(hyp² − leg₁²), the identity a² + b² = c² worked by subtraction rather than addition. A tool that adds two known legs to build a hypotenuse only ever produces a bigger number; this one runs the opposite way, peeling a known square off the longest side to see what square is left underneath.
Whole-number answers like the one below are not accidents. Pick any two whole numbers m > n > 0 and build a triple from m² − n², 2mn, and m² + n² — that trio always satisfies a² + b² = c², a generating rule preserved in Book X of Euclid's Elements. Set m = 3 and n = 2: the legs come out to 9 − 4 = 5 and 2 × 3 × 2 = 12, and the hypotenuse to 9 + 4 = 13 — the exact 5-12-13 triangle this page solves, produced from nothing but two small integers and one piece of algebra.
Addition never fails, but this subtraction can. Because the answer is a square root, hyp² − leg₁² cannot drop below zero, which means the known side can never reach or pass the hypotenuse — a right triangle's longest side is always the one facing the right angle, never one of the other two. Push Known leg up toward Hypotenuse and Missing leg shrinks toward zero; let the two become equal and the triangle has already flattened into a line before the square root has anything left to return, so this sheet flags that pair instead of guessing at one.
- Enter the triangle's longest side into Hypotenuse, the measurement that stretches across from the right angle.
- Enter the side length you already have into Known leg — either of the two shorter sides forming that right-angle corner.
- Read the answer straight from Missing leg; it recalculates the instant either input changes.
- Keep Hypotenuse and Known leg in one consistent unit — Missing leg returns in that identical unit.
- If Known leg is set equal to or longer than Hypotenuse, the sheet flags the pair instead of returning a result, since no such triangle exists.
Worked example — a 13-foot ladder
A 13-foot ladder leans against a wall with its base set 5 feet out from it. Enter Hypotenuse = 13 and Known leg = 5: Missing leg = √(13² − 5²) = √(169 − 25) = √144 = 12, exactly. The ladder reaches 12 feet up the wall — the 5-12-13 triple, recovered here from a hypotenuse and one leg rather than assembled from two legs the way a forward Pythagorean solve would build it.
Double both known measurements and the answer scales right along with them: Hypotenuse = 26 and Known leg = 10 gives Missing leg = √(676 − 100) = √576 = 24 — the same 5-12-13 triangle at twice the size, since similar right triangles keep every ratio between their sides fixed no matter how far you stretch them.
Questions
What is the formula to find a missing leg of a right triangle?
leg₂ = √(hyp² − leg₁²): square the hypotenuse, subtract the square of the leg you know, then take the square root. It is the Pythagorean theorem a² + b² = c² solved for a leg instead of the hypotenuse, so hyp = 13 and leg₁ = 5 gives leg₂ = √(169 − 25) = √144 = 12.
What separates this sheet from one that solves for the hypotenuse?
That one adds two known legs, c = √(a² + b²), and the result always comes out longer than either input. This sheet subtracts, leg₂ = √(hyp² − leg₁²), and the result always comes out shorter than the hypotenuse — the two tools solve the identical triangle by running the theorem in opposite directions.
What happens if the known leg is not shorter than the hypotenuse?
The sheet flags the pair instead of returning a number, because hyp² − leg₁² would be zero or negative, and no real length squares to something negative. A right triangle's leg is always shorter than its hypotenuse, so reaching that limit means the two figures you entered cannot both belong to the same triangle — the check catches a data-entry slip before it produces a fake answer.
Where do whole-number answers like 5, 12, and 13 come from?
From Euclid's rule for generating them: pick whole numbers m > n > 0 and build m² − n², 2mn, and m² + n². With m = 3 and n = 2 that yields 5, 12, and 13 exactly — the algebra behind every clean right triangle this sheet returns, recorded in Book X of Euclid's Elements over two thousand years ago.
Does the missing leg scale if I scale the other two numbers?
Yes, proportionally. Multiply Hypotenuse and Known leg by the same factor and Missing leg is multiplied by it too — scale 13 and 5 up to 26 and 10, and the answer doubles from 12 to 24. Similar right triangles keep identical angle and side ratios at any size.
Can the missing leg come out longer than the known leg I entered?
Yes — which one is longer depends entirely on the numbers, not on which side you happened to measure first. With Hypotenuse = 13 and Known leg = 5, the missing leg of 12 is longer than the known one; enter Known leg = 12 instead and Missing leg returns 5, the same triangle read from its other side.