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Legs 3 and 4 → hypotenuse exactly 5. Hypotenuse 13 with leg 5 → the other leg is 12. The theorem carpenters trust more than their squares.
Two thousand five hundred years old and still the hardest-working theorem in math class and on the job site: a² + b² = c² for any right triangle. This calculator solves both directions — two legs into a hypotenuse, or hypotenuse-and-leg into the missing leg — and flags when your numbers land on a perfect integer triple like 3-4-5 or 8-15-17.
Students and carpenters use the same theorem: legs 3 and 4 → hypotenuse exactly 5. Solves for hypotenuse or missing leg, and flags the perfect triples.
Triples are whole-number side sets that satisfy a² + b² = c² exactly: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and 20-21-29 are the classics. They matter beyond homework because they give exact right angles with no measurement error — carpenters square foundations and deck corners with the 3-4-5 rule: mark 3 ft on one side and 4 ft on the other, and when the diagonal reads exactly 5 ft the corner is perfectly 90°. Scaling works too: 6-8-10 and 9-12-15 are the same triple multiplied. Any multiple of any triple is another triple, so 3-4-5 alone covers everything from picture frames to house slabs. The generator formula (m²−n², 2mn, m²+n² for m>n) produces every primitive triple — m=2, n=1 gives 3-4-5.
Anywhere a diagonal matters. Construction: squaring corners with 3-4-5, sizing rafters (run² + rise² = rafter²), and checking that a rectangular foundation is true by comparing its two diagonals. Navigation and logistics: straight-line distance between two points is the hypotenuse of the east-west and north-south legs — 30 miles east and 40 miles north is exactly 50 miles direct. Screen sizes: a "55-inch TV" is the diagonal — a 48×27-inch screen is 55.1 inches diagonal. Sports: the throw from third to first is the hypotenuse of a 90-ft diamond (127.3 ft). The theorem also hides inside every distance formula in physics, statistics (standard deviation is a generalized hypotenuse), and machine learning (Euclidean distance).
No — a² + b² = c² is exactly true only for right triangles, and the deviation from equality actually classifies the triangle: if c² < a² + b² the triangle is acute (all angles under 90°), if c² > a² + b² it is obtuse. That comparison is the fastest way to classify a triangle from side lengths alone. For non-right triangles the generalization is the Law of Cosines: c² = a² + b² − 2ab·cos(C), which reduces to Pythagoras when C = 90° because cos(90°) = 0 — the theorem is the special case, not the exception. One more caveat: the theorem assumes flat (Euclidean) geometry, which is why long-distance flight paths on a sphere use great-circle formulas instead.