Find a right triangle’s hypotenuse — with the full Pythagorean working shown.
Three-four-five — the most famous triangle in history: legs 3 and 4, hypotenuse exactly 5. The hypotenuse is the side opposite the right angle, and the Pythagorean theorem (a² + b² = c²) finds it from the two legs. Our free Hypotenuse Calculator computes c = √(a²+b²), shows every step, and reports the acute angles plus triple detection.
Why the hypotenuse matters: it is the shortest distance between two points’ diagonal — the ladder’s length, the screen’s diagonal, the flight’s direct route. Builders use 3-4-5 triangles to square foundations; navigators use the theorem on every chart; video specs quote the hypotenuse (a “55-inch TV” measures diagonally).
Enter the two legs and the calculator returns the hypotenuse to two decimals with the full substitution, the two acute angles, and a Pythagorean-triple verdict. Whether you are framing a wall, buying a TV, or doing geometry homework, the steps are shown and the math is exact.
In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². The hypotenuse (from Greek hypoteinousa, “stretching under”) is the longest side, opposite the 90° angle. The theorem is attributed to Pythagoras (c. 570–495 BC) but was known to Babylonians a millennium earlier.
Pythagorean triples — whole-number solutions like (3,4,5), (5,12,13), (8,15,17), (7,24,25) — are the theorem’s celebrities. Multiples work too: (6,8,10) is just 2×(3,4,5). The widget flags triples because they let you skip the root entirely.
The converse is equally useful: if a² + b² = c², the angle is right. Builders “square” foundations with a 3-4-5 rope — the converse in work boots.
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The widget also reports the acute angles via arctan — the full triangle solution from just two sides.
Find the hypotenuse of legs 3 and 4.
The triple every builder memorizes: 3-4-5 squares a corner with a tape measure and no math.
A ladder’s foot is 5 ft from a wall; it reaches 12 ft up. How long is the ladder?
Another triple — which is why ladder problems so often come out clean: textbook authors love (5,12,13).
Jonah is laying out a deck foundation and needs one corner perfectly square. He measures 3 ft along one string, 4 ft along the other, and adjusts until the diagonal reads exactly 5 ft — the 3-4-5 triple, no calculator needed. The widget confirms: √(9+16) = 5. The deck comes out square because the converse guarantees it.
A second scenario: a “55-inch TV” — that’s the diagonal. With a 16:9 screen, width = 48 in, height = 27 in: √(48²+27²) = √(2304+729) = √3033 = 55.07 — the marketing number, derived.
A third: a hiker walks 3 km east then 4 km north. The direct return is 5 km — the hypotenuse shortcut vs. 7 km back along the trail. Pythagoras plans the shortcut.
A fourth: a 10-ft ladder must reach a 9-ft roof edge; how far out is the foot? a = √(100−81) = √19 = 4.36 ft — the rearranged formula (leg from hypotenuse) the widget’s note covers.
Construction: the 3-4-5 rope stretches from Egyptian pyramid builders to modern framers — the oldest applied mathematics still in daily use. “Squaring” a foundation is the converse of Pythagoras in action.
Navigation: dead reckoning resolves north/east legs into the direct-track hypotenuse; GPS receivers solve the same geometry in three dimensions. Every “as the crow flies” distance is a hypotenuse.
Technology: screen diagonals define product sizes (phones, TVs, monitors); the 16:9 aspect’s diagonal follows √(16²+9²) = √337 per unit. Pixel density (PPI) divides the diagonal pixels by the diagonal inches.
Physics: vector magnitudes are hypotenuses — a 3 N east + 4 N north force is 5 N northeast. Resultant vectors, phasors, and error propagation all root-sum-square.
Statistics: standard deviation of independent errors combines by root-sum-of-squares — Pythagoras in error space. The 3-4-5 of uncertainty budgets.
Number theory: triples parametrize via Euclid’s formula (m²−n², 2mn, m²+n²); Fermat’s Last Theorem says no cube analog exists. The hypotenuse launched a 350-year mathematical quest.
Art and design: the √2 rectangle (A-series paper) comes from the unit square’s diagonal — the hypotenuse as a design standard (ISO 216).
One pattern worth internalizing: root-sum-of-squares appears wherever independent perpendicular contributions combine — geometry, vectors, errors, signals. Recognizing the pattern turns a triangle formula into a universal tool.
Map reading: 3 km east and 4 km north is 5 km straight-line — the hypotenuse is the “as the crow flies” distance. Hikers estimate shortcuts with 3-4-5 intuition before touching a calculator.
Roofing: a 6:12 roof pitch rises 6 per 12 run; the rafter length per foot of run is √(12²+6²)/12 = √180/12 = 1.118 ft. Roofers multiply by 1.118 reflexively — the hypotenuse as a trade constant.
Sailing: beating upwind at 45°, velocity-made-good is the hypotenuse’s leg: 6 knots boat speed × cos45° = 4.24 knots toward the mark. Every tack is a right triangle.
Data science: Euclidean distance between points is the hypotenuse generalized — √(ΣΔ²). Clustering, nearest-neighbors, and embeddings all measure “distance” as multi-dimensional hypotenuses.Wilderness navigation: 3 km east then 4 km north leaves you 5 km from camp in a straight line — the hypotenuse every hiker estimates before deciding whether to bushwhack back or retrace the trail. Map-and-compass courses teach the 3-4-5 estimate deliberately: 6 km east and 8 km north gives 10, and 5 and 12 gives 13, because the classic triples make the mental math trivial. The real skill is judging when the shortcut pays — 5 km through dense forest at 2 km/h is slower than 7 km of trail at 4 km/h — and the widget’s exact figure feeds that decision instead of replacing it.
Roof framing: a 6:12 pitch rises 6 inches per 12 inches of run, so each foot of horizontal run needs √(12²+6²)/12 = √180/12 = 1.1180 ft of rafter. Carpenters multiply the building’s half-span by this “rafter factor” reflexively — a 14-ft half-span needs 15.65 ft of rafter before ridge and eave allowances. The factor table on every framing square is just precomputed hypotenuses: 1.4142 for 12:12 pitch (√2), 1.1180 for 6:12, 1.0308 for 3:12. The widget derives any pitch’s factor on demand — which is all the framing square ever was, a hypotenuse lookup table.
Machine learning: Euclidean distance — √(Σ(xi−yi)²) — is the hypotenuse generalized to n dimensions, underlying k-nearest-neighbors, k-means clustering, and embedding similarity. A recommendation engine calling two users “similar” has computed a hypotenuse through thousand-dimensional taste space. The curse of dimensionality is a hypotenuse phenomenon: as dimensions grow, all pairwise distances converge and the nearest neighbor stops being meaningfully near. Every “people like you also bought” is Pythagoras, iterated ten thousand times per query.
Emergency dispatch: a fire truck 4 blocks east and 3 blocks north of a call is 5 blocks away straight-line — dispatchers estimate response distance with 3-4-5 intuition before the GPS refines it. In a grid city the hypotenuse beats the Manhattan route whenever diagonal shortcuts exist, and the closest-station algorithm is a nearest-hypotenuse search across the whole fleet.
Hypotenuse: c = sqrt(a^2 + b^2). Triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Converse squares corners. Source: Sigma Calculator, Hypotenuse Calculator, https://sigmacalculator.net/math-calculators/hypotenuse-calculator/
Cite this page as: Hypotenuse Calculator, Sigma Calculator, https://sigmacalculator.net/math-calculators/hypotenuse-calculator/
Last updated: October 12, 2026