Give any two sides of a right triangle and get the third, plus its area and perimeter.
Legs 3 and 4 give a hypotenuse of 5 — but this page also quietly computes two figures no FAQ below mentions: an area of ½ × 3 × 4 = 6, and a perimeter of 3 + 4 + 5 = 12. Both fall straight out of the same three side lengths once the hypotenuse is known.
Arrange four copies of the same right triangle around a tilted inner square of side c, forming one large outer square of side (a + b). The outer square's area is (a + b)², and it is also exactly the four triangles plus the inner square: 4 × (½ab) + c². Setting those equal — (a + b)² = 2ab + c² — and expanding the left side gives a² + 2ab + b² = 2ab + c², and the 2ab cancels from both sides, leaving a² + b² = c². No trigonometry, no limits — just two ways of measuring the same square.
Try to solve for a leg with a hypotenuse of 3 and the other leg at 4, and this page refuses rather than returning a number — because a leg can never exceed the hypotenuse. The hypotenuse is defined as the longest side of a right triangle, so if the "hypotenuse" entered is shorter than a leg, the two numbers cannot describe a real right triangle at all; there is no length that would make a² + b² = c² balance. The same 3-4-5 relationship appears as coordinates on the Coordinate Distance Calculator and as a cone's radius, height, and slant edge on the Volume Calculator — the theorem is the same regardless of which two quantities it's connecting.
In any right triangle the square of the hypotenuse equals the sum of the squares of the other two sides. Rearranged for each unknown:
The hypotenuse is always the longest side and always opposite the right angle. If you are solving for a leg and the "hypotenuse" you entered is shorter than the other leg, no such triangle exists — the calculator says so rather than returning a nonsense number.
Some right triangles have whole numbers on all three sides. The smallest is 3, 4, 5, and any multiple of it works too — 6, 8, 10 and 9, 12, 15 are the same triangle scaled up. Other primitive triples include 5-12-13, 8-15-17 and 7-24-25.
Builders use the 3-4-5 relationship to check a corner is square without a set square: measure 3 units along one edge, 4 along the other, and if the diagonal is exactly 5 the angle is 90°.
This is worth stating because the theorem gets applied where it does not belong. For a triangle without a right angle you need the law of cosines, c² = a² + b² − 2ab·cos(C), which reduces to Pythagoras exactly when C is 90° and cos(C) is zero. The Triangle Calculator handles that case.
It is the side opposite the right angle, and always the longest of the three. If a triangle has no right angle it has no hypotenuse at all.
3-4-5 is the smallest, followed by 5-12-13, 8-15-17 and 7-24-25. Any whole-number multiple of a triple is also a triple, so 6-8-10 and 30-40-50 both work.
Most likely the hypotenuse you entered is shorter than or equal to one of the legs, which is geometrically impossible. The hypotenuse must be the longest side.
No. Use the law of cosines instead, which the Triangle Calculator applies. Pythagoras is the special case of that law where the angle is exactly 90°.
For a right triangle specifically — not just the missing side, but both acute angles, the area, and the perimeter together — the Right Triangle Calculator solves the whole shape from the same two knowns.
For right triangles where you know an angle rather than two sides, use the Right Triangle Calculator; for triangles with no right angle, the Triangle Calculator. The distance formula is this theorem applied to coordinates — see the Coordinate Distance Calculator. Two practical applications on the site: the Roofing Calculator uses it to derive the pitch multiplier, and the Stair Calculator uses it for stringer length.