How this instrument works
A right triangle carries less freedom than it first appears to. Fix the two legs, Leg a and Leg b, and the shape is completely decided: they already sit at 90° to each other by definition, and two sides plus the fixed angle between them is exactly the side-angle-side condition that pins a triangle down to one and only one possible shape. Every other measurement — the third side from a² + b² = c², the enclosed area as half the legs' product, the boundary length as their sum with that third side added on, and the pair of remaining acute angles from the ratio of one leg to the other — is not new information so much as a forced consequence of the two numbers already entered.
This page exists because those forced consequences are usually asked about one at a time elsewhere on this site: a hypotenuse by itself, an area by itself, a perimeter by itself, an angle by itself, each behind its own narrower panel built around the same underlying triangle. Here, the identical two legs answer all four questions in one pass, which suits a real object that needs more than one figure at once — a roof truss wanting both its rafter length and its pitch, say — since re-entering the same two measurements four separate times only invites a mismatched typo on the third try.
Two edges of the formula are worth knowing. Set both legs to 1 and the isosceles case appears: the hypotenuse works out to exactly √2, the area to 0.5, the perimeter to 2 + √2, and both acute angles lock at precisely 45°, the single leg pairing whose two non-right angles can never be told apart. A stranger coincidence turns up at legs 5 and 12: Area comes back as 30, and Perimeter also comes back as 30 — 0.5 × 5 × 12 and 5 + 12 + 13 both happen to total the same number, even though one measures a surface in square units and the other a boundary in linear units. Nudge either leg away from 5 or 12 by even a fraction and the two totals split apart immediately; it is a coincidence of that one pair, not a rule.
- Enter your triangle's two legs into Leg a and Leg b, the pair that frames the 90° corner, using one consistent unit for both.
- Read Hypotenuse for the third side, found from a² + b² = c² the moment both legs are set.
- Read Area and Perimeter alongside it; both come from the same two legs with no extra entry required.
- Read Angle opposite a for the acute angle across from Leg a; it displays in degrees unless you pick radians or turns from that field's own unit selector.
- Swap the values in Leg a and Leg b to read the triangle's OTHER acute angle in Angle opposite a, since the field always answers relative to whichever length currently sits in Leg a.
Worked example — a shelf bracket built from 3 and 4 unit legs
A steel shelf bracket is cut as a right triangle to brace a shelf against a wall, with the wall-side leg (Leg a) measuring 3 units and the shelf-side leg (Leg b) measuring 4 units, meeting at the fixed 90° corner where the bracket bolts to both surfaces. Entering only those two lengths fills every other field at once: Hypotenuse = √(3² + 4²) = √25 = 5.0, the length to cut for the diagonal brace; Area = ½ × 3 × 4 = 6.0, the sheet metal the bracket consumes; Perimeter = 3 + 4 + 5 = 12.0, the total edge a fabricator needs to deburr; and Angle opposite a = atan(3 ⁄ 4) = 0.6435011087932844 radians, reported by default as 36.86989764584402°, about 36.87°, the tilt of the diagonal brace measured from the shelf-side leg.
Change only Leg b to 12 and keep Leg a at 3, and all four fields move together rather than one at a time: Hypotenuse becomes roughly 12.3693, Area drops to 18.0, Perimeter rises to about 27.3693, and Angle opposite a narrows to about 14.04°. One edit updates every figure instead of four separate lookups across the site's narrower hypotenuse, area, perimeter, and angle calculators, because a right triangle's two legs already carry everything else needed to describe it.
Questions
Why do just two legs solve the whole triangle?
Because the angle between them is fixed at 90° by definition, the two legs satisfy the side-angle-side condition that pins down one unique triangle. Every other measurement — hypotenuse, area, perimeter, both acute angles — is then a forced consequence of those two numbers, not an independent fact needing its own measurement.
What does this add beyond the site's separate hypotenuse, area, perimeter, and angle calculators?
Those each take the same two legs and hand back a single figure. This instrument runs all four relationships — c = √(a²+b²), A = ½ab, P = a+b+c, and ∠A = atan(a⁄b) — on one entry of Leg a and Leg b, so a project needing more than one of those numbers skips re-entering the same two lengths repeatedly.
Does Angle opposite a mean the right angle itself?
No — the right angle is fixed at 90° and is not one of the outputs here. Angle opposite a is the acute angle across from Leg a, at the vertex where Leg b meets the hypotenuse. Swap the values in Leg a and Leg b to read the triangle's other acute angle in that same field.
Can the four results ever disagree with each other?
No — all four are read off the same fixed triangle, so they cannot contradict one another. Perimeter always equals Leg a + Leg b + Hypotenuse exactly, and Area always equals half of Leg a times Leg b exactly; a hand check that disagrees with either sum points to an arithmetic slip, not to the triangle.
Does it matter if I accidentally type the hypotenuse where a leg should go?
Leg a and Leg b must both be the triangle's legs, not the hypotenuse. Typing the hypotenuse into either field describes a larger, different triangle, so every downstream figure — area, perimeter, angle — would answer for the wrong shape entirely. If the hypotenuse is one of your two known lengths, a dedicated missing-side tool elsewhere on this site recovers the other leg first.
Why does Area equal Perimeter for legs of 5 and 12?
It is a coincidence specific to that one pair, not a rule: 0.5 × 5 × 12 and 5 + 12 + 13 both happen to total 30, even though one is a square-unit area and the other a linear-unit length. Nudge either leg away from 5 or 12 and the two totals separate immediately — a handy one-off check, not a pattern to expect elsewhere.