How this instrument works
A right triangle's perimeter is nothing more than three sides added together, but two legs alone are enough to name all three, because the right angle between them fixes the third side automatically. Substitute c = √(a² + b²) for the hypotenuse directly into the ordinary perimeter sum P = a + b + c, and the two steps collapse into one expression: P = a + b + √(a² + b²). No angle needs measuring and no diagonal needs a separate tape reading — the 90° corner already supplies it.
This page assumes exactly one starting point: you know the two legs, the sides meeting at the right angle, not any other pair. That differs from a rectangle's perimeter, built from length and width with no square root anywhere (twice their sum, since a rectangle has four sides in two matching pairs), or from a rectangle's perimeter recovered from its area and one side, which needs a division first. Legs are usually the easiest pair to read directly off a square-cornered object — a ramp's rise and run, a roof truss's vertical and horizontal timbers — while the hypotenuse is the awkward diagonal nobody has a straight edge against. Hand this page the legs and it supplies the diagonal itself before totalling all three sides.
One boundary case is worth knowing: shrink a leg to zero and the triangle collapses flat, doubling back on itself; the surviving leg becomes the hypotenuse too, so the perimeter reduces to exactly twice its length rather than needing special handling. At the other extreme, the perimeter always lands strictly between twice the longer leg and twice the sum of both legs — the hypotenuse is always longer than either leg alone, since it sits opposite the right angle, yet always shorter than their sum, the ordinary triangle inequality, and both limits show up directly in the total.
- Enter the first known measurement into Leg a — either of the two sides meeting at the right angle, in any consistent unit.
- Enter the second measurement into Leg b, using that same unit.
- Read Perimeter — the total distance around the triangle, with the hypotenuse already found and added in behind the scenes.
- Leg a and Leg b are interchangeable; addition does not care which one you call which, so the result is identical either way round.
Worked example — fencing a 3-4 right-triangle plot
A triangular corner of garden is bounded by two straight fences meeting at a right angle: Leg a = 3 metres and Leg b = 4 metres. The instrument first recovers the missing third side the way any right triangle demands: c = √(3² + 4²) = √25 = 5 metres, the boundary running diagonally back to close off the plot.
Perimeter then sums all three: P = 3 + 4 + 5 = 12 metres of edging to order, with nothing left over and nothing short. Double every measurement instead — legs of 6 and 8 — and the whole triangle scales up together, hypotenuse included, for a perimeter of exactly 24 metres, because every term in P = a + b + √(a² + b²) grows by the same factor when a and b do.
Questions
What is the formula for the perimeter of a right triangle?
P = a + b + √(a² + b²), where a and b are the two legs meeting at the right angle. The square-root term recovers the hypotenuse via the Pythagorean theorem before all three sides are added, so a right triangle's perimeter needs only two measurements instead of the three a general triangle requires.
How is this different from finding a triangle's perimeter from three sides?
A general triangle's perimeter is simply P = a + b + c, and all three sides must be measured independently because nothing ties them together. A right triangle removes that need: its 90° angle links the hypotenuse to the two legs by c = √(a² + b²), so only the legs are required and the third side is calculated rather than measured.
Why does the calculator ask for the two legs instead of the hypotenuse?
Because the legs are usually the pair that's easiest to read directly off a square-cornered object — a rise and a run, a base and a height — while the hypotenuse is the diagonal line nobody has a straight edge against. Given the legs, the Pythagorean theorem supplies the hypotenuse without a second physical measurement, so two numbers are always enough.
What happens if one leg is zero?
The triangle collapses into a straight line doubling back on itself: with Leg a = 0, the hypotenuse equals Leg b exactly, and the perimeter reduces to 2 × b. Entering a = 0 and b = 4, for instance, returns a perimeter of 8 rather than an error, since a degenerate triangle is still a well-defined limiting case of the formula.
What is a common mistake when applying this formula?
Adding the two legs and doubling the result, giving P = 2(a + b) instead of a + b + √(a² + b²). The legs never sum to the hypotenuse except in the degenerate case where one leg is zero; for any genuine right triangle the hypotenuse is strictly shorter than a + b, so that shortcut always overstates the true perimeter.
Does doubling both legs double the perimeter?
Yes — the formula is homogeneous of degree one: scale both legs by any factor k and every term, including the square root, scales by that same k, so the perimeter scales by k too. Legs of 6 and 8, double 3 and 4, give a perimeter of 24, exactly double the 12 you get from the original pair.