How this instrument works
This instrument answers one specific question: given the two legs of a right triangle, how long is the side opposite the right angle? Leg a and Leg b go in; Hypotenuse comes out as c = √(a² + b²), with no reverse mode to switch on, since the direction is fixed. The name itself comes from the Greek hypoteinousa, 'the side stretching under' the right angle — a description that still matches the picture, since tilting a right triangle onto its point leaves the hypotenuse as the one side lying underneath, spanning between the two legs' far ends.
One proof of c² = a² + b² needs no trigonometry at all, only rearrangement. Draw a square with side length (a + b) and tile it two ways. Arrange four copies of the triangle around the inside edge, tilted so their hypotenuses trace a smaller square in the middle — that leftover square has area c². Slide the same four triangles into the corners instead, and the leftover space splits into two separate squares, of side a and side b, with combined area a² + b². Both arrangements start from the identical outer square and use four identical triangles, so the leftover areas must match, and c² = a² + b² falls out with no algebra needed.
Two limits are worth knowing. Send one leg to zero and the triangle flattens into a straight segment, so the hypotenuse simply equals whichever leg survived — a boundary this instrument still computes correctly rather than rejecting. Set both legs to exactly 1 instead, and the hypotenuse comes back as √2 ≈ 1.41421356, a number with no exact fraction anywhere in its decimal expansion. That particular length is the one credited with convincing the Pythagoreans that whole-number ratios cannot describe every length — an unsettling discovery made with the plainest right triangle there is.
- Enter the length of the first leg into Leg a — one of the two sides forming the right angle.
- Enter the length of the second leg into Leg b — the other side forming that same right angle.
- Read the answer straight from Hypotenuse; it recalculates the moment either leg changes.
- Keep both legs in one consistent unit, inches with inches or metres with metres, since Hypotenuse returns in that same unit.
Worked example — legs of 3 and 4
Enter Leg a = 3 and Leg b = 4. Squaring each leg gives 9 and 16; adding them gives 25; taking the square root gives Hypotenuse = √25 = 5, exactly, with no digits left over to round away. This is the smallest right triangle with three whole-number sides, but the instrument reaches c = 5 the same way it reaches any other answer: square, add, root, with no shortcut and no special-casing for a tidy result.
The same three steps hold at both edges of the input range. Set Leg a = 0 and Leg b = 5, and the triangle collapses to a line: Hypotenuse returns exactly 5, matching the surviving leg with nothing left to compute. Set Leg a = 1 and Leg b = 1 instead, and Hypotenuse returns 1.41421356 — the diagonal of a unit square, correct to eight decimal places, an irrational answer produced by two perfectly ordinary whole-number legs.
Questions
What is the formula for the hypotenuse of a right triangle?
c = √(a² + b²), where a and b are the two legs meeting at the right angle. Square each leg, add the squares, then take the square root — there is no shortcut that skips the squaring step, and no simpler exact formula exists for a general right triangle.
Why can't I just add the two legs to get the hypotenuse?
Because length does not combine that way once a right angle is involved. For legs 3 and 4, a + b = 7, but the true hypotenuse is 5; plain addition ignores the angle between the legs entirely, and only squaring, summing, and rooting accounts for it correctly.
Is the hypotenuse always the longest side of the triangle?
Yes. Since c² = a² + b² and both squared terms are positive whenever a leg is nonzero, c² must exceed either a² or b² alone, so c itself exceeds either leg on its own. The hypotenuse can equal a leg only in the degenerate case where the other leg is zero.
How does this connect to the distance formula on a coordinate plane?
The distance formula is this same calculation under a different name. Two points separated by Δx horizontally and Δy vertically sit at the ends of a right triangle's hypotenuse, so distance = √(Δx² + Δy²) is c = √(a² + b²) with the legs relabelled as coordinate differences.
Why does √2 keep showing up when both legs equal 1?
Because a unit right triangle's hypotenuse is irrational: √2 ≈ 1.41421356, a value with no exact fraction however far the decimals run. That case is historically significant — it is the length that first showed the Pythagoreans that whole-number ratios cannot describe every length, even one built from two perfectly whole legs.
What happens if I set one leg to zero?
The triangle collapses into a straight line, and Hypotenuse simply equals whatever the surviving leg measures — set Leg a to 0 and Leg b to 5, and Hypotenuse returns exactly 5. It is the boundary where the shape stops being a triangle, and the formula keeps working right up to it.