Find the hypotenuse of a right triangle. Choose what you know — two legs, a leg and an angle, the area and a leg, or a leg and the hypotenuse — and the calculator returns the hypotenuse along with the remaining sides and angles. The right angle is at C, so c is the hypotenuse, a and b are the legs, and α and β are the acute angles opposite a and b.
α is the angle opposite leg a.
The hypotenuse must be longer than the leg.
The hypotenuse calculator finds the hypotenuse — the longest side of a right triangle, the one opposite the right angle — from whatever information you have. Give it the two legs, a leg and an angle, the area and a leg, or a leg and the hypotenuse itself, and it returns the hypotenuse together with the other sides and both acute angles. It is the fast, reliable way to solve the "how long is the slanted side?" problem that comes up in geometry class, carpentry, construction, and design.
Every right triangle has one 90° angle, two shorter sides called legs, and one long side called the hypotenuse. Because those parts are locked together by the Pythagorean theorem and the basic trig ratios, knowing just two of them is enough to find all the rest — which is exactly what this calculator does, in four different ways.
The hypotenuse is the side of a right triangle that sits directly opposite the right angle. It is always the longest of the three sides, because the right angle is the largest angle in the triangle and the longest side always faces the largest angle. The other two sides, which meet at the right angle, are called the legs (or catheti). In this calculator the right angle is at vertex C, so the hypotenuse is labeled c, and the legs are a and b.
Finding the hypotenuse is one of the most common tasks in geometry because it turns a horizontal distance and a vertical distance into a single straight-line distance — the diagonal. Whenever you need the length of a ramp, a rafter, a ladder against a wall, or the diagonal of a rectangle, you are finding a hypotenuse.
The classic way to find the hypotenuse from the two legs is the Pythagorean theorem:
c = √(a² + b²)
In words: square each leg, add them together, and take the square root. For the famous 3-4-5 triangle, c = √(3² + 4²) = √(9 + 16) = √25 = 5. But the two legs are not the only starting point. This calculator also finds the hypotenuse when you know:
Whichever combination you have, the calculator picks the right formula automatically and reports every side and angle of the triangle.
Two legs. A rectangle is 3 m by 4 m; how long is its diagonal? Enter legs a = 3 and b = 4: the hypotenuse is √(9 + 16) = 5 m, and the acute angles are about 36.87° and 53.13°.
Leg and angle. A ladder leans so that its foot is 10 ft from the base of a wall and the ladder makes a 75.5° angle at the top with... using leg a = 10 and α = 75.5°: the hypotenuse (the ladder) is c = 10 ÷ sin(75.5°) ≈ 10.33 ft, the other leg is about 2.586 ft, and β = 14.5°.
Leg and hypotenuse (reverse). If one leg is 12 cm and the hypotenuse is 13 cm, the missing leg is √(13² − 12²) = √(169 − 144) = √25 = 5 cm — the well-known 5-12-13 right triangle.
Because it covers four input combinations, the tool works as a hypotenuse calculator, a missing-leg calculator, and a right-triangle solver all at once.
The relationship c² = a² + b² is one of the oldest and most important results in mathematics, attributed to the ancient Greek mathematician Pythagoras, though it was known to earlier civilizations. It says that the area of the square built on the hypotenuse equals the combined area of the squares built on the two legs. This is true for every right triangle, and only for right triangles — which is why the theorem is also used in reverse to test whether a triangle has a right angle at all. If three side lengths satisfy a² + b² = c², the triangle must be right-angled. Builders use this "3-4-5 rule" to square up corners: measure 3 units along one edge, 4 along the other, and if the diagonal is exactly 5, the corner is a perfect right angle.
A Pythagorean triple is a set of three whole numbers that fit the Pythagorean theorem exactly, giving a right triangle with all-integer sides. The most famous are 3-4-5 and 5-12-13, both of which appear in the examples above. Others include 8-15-17, 7-24-25, and 20-21-29, plus every multiple of a triple (6-8-10, 9-12-15, and so on). These triples are prized in construction and design because they produce clean right angles without any messy decimals. When your calculator result comes out as a whole number, you have likely stumbled onto a Pythagorean triple — and this tool makes it easy to spot them.
When you know an angle instead of the second leg, trigonometry provides the link. In a right triangle, the sine of an acute angle equals the opposite side divided by the hypotenuse, so sin(α) = a / c, which rearranges to c = a / sin(α). Likewise, cosine relates the adjacent leg to the hypotenuse, cos(α) = b / c. These ratios are why the "Leg & Angle" mode works: one side and one acute angle completely determine a right triangle. The calculator uses these trig relationships behind the scenes, and it also reports both acute angles for you, remembering that in any right triangle the two non-right angles always add up to 90°.
The word hypotenuse comes from the ancient Greek hypoteinousa, meaning "stretching under" — a fitting description of the long side that stretches beneath the right angle, connecting the ends of the two legs. The legs themselves are sometimes called catheti (singular cathetus), from the Greek for "let down" or "perpendicular." These terms are more than trivia: they encode the geometry. The hypotenuse is defined entirely by its relationship to the right angle, which is why the concept only exists for right triangles. In a triangle with no 90° angle, there is no hypotenuse at all — just three ordinary sides. Understanding this is the key to knowing when the tools on this page apply: they work precisely because a right angle guarantees the clean relationships (the Pythagorean theorem and the sine, cosine, and tangent ratios) that make the hypotenuse so easy to find. Whenever you can identify a right angle in a problem, you can find its hypotenuse.
A few right triangles show up so often that their hypotenuses are worth memorizing. For the "special" triangles defined by their angles:
And for the integer-sided "Pythagorean triple" triangles, the hypotenuse is always a whole number:
Recognizing these lets you check the calculator's output at a glance and speeds up mental estimates on the job site or in an exam.
A frequent real-world situation is knowing only one leg plus some other piece of information — not both legs. That is exactly why this calculator offers four modes. If you know one leg and an angle, the "Leg & Angle" mode uses trigonometry (c = leg ÷ sin of the opposite angle) to find the hypotenuse directly. If you know the triangle's area and one leg, the "Area & Leg" mode recovers the second leg from the area formula (area = ½ × base × height, so the other leg is 2 × area ÷ the known leg) and then applies the Pythagorean theorem. And if you already know the hypotenuse and want a missing leg, the "Leg & Hypotenuse" mode rearranges the theorem to b = √(c² − a²). In every case, one leg on its own is never enough — you always need a second independent fact about the triangle — but with any such pair, the calculator finds the hypotenuse and completes the triangle. This flexibility is what makes it useful beyond the textbook, where you rarely happen to have both legs handed to you and often must work from an angle of elevation, a known area, or a measured diagonal instead.
Square both legs, add them, and take the square root: c = √(a² + b²). For legs of 3 and 4, the hypotenuse is √25 = 5. If you know a leg and an angle instead, use c = a ÷ sin(α).
Yes. Rearrange the Pythagorean theorem: b = √(c² − a²). For a hypotenuse of 13 and a leg of 12, the other leg is √(169 − 144) = 5. Use the "Leg & Hypotenuse" tab.
Yes. The hypotenuse is opposite the right angle, which is the largest angle in a right triangle, so it is always longer than either leg.
One side alone is not enough — you also need an angle or the area. With a leg and an acute angle, use c = a ÷ sin(α); with the area and a leg, first find the second leg (b = 2 × area ÷ a), then apply the Pythagorean theorem.
It is a set of three whole numbers that satisfy a² + b² = c², such as 3-4-5 or 5-12-13. They give right triangles with all-integer sides, which is handy in construction and design.
No — only on right triangles. For triangles without a 90° angle, use the law of cosines, c² = a² + b² − 2ab·cos(C), which reduces to the Pythagorean theorem when C is 90°.
This Hypotenuse Calculator is provided for educational and general informational purposes. It uses the standard right-triangle formulas; verify results independently for critical engineering, construction, or academic work.