About the Theorem
The most famous equation in geometry earns its fame by showing up everywhere: ladder lengths, TV diagonals, stair stringers, plot diagonals, navigation legs, and the distance formula itself (which is this theorem wearing coordinates). Twenty-five centuries old and still on every job site.
Two modes: enter both legs for the hypotenuse, or a leg plus the hypotenuse for the missing leg. Results include the triangle's area, perimeter, and acute angles — and when your numbers form a perfect triple (3-4-5 and family), the answer lands exact.
The coordinate-grid version of the same idea is the Slope Calculator.
Both Directions
One theorem, two rearrangements:
Hypotenuse: c = √(a² + b²) Missing leg: a = √(c² − b²) (c must exceed b) Angles: atan(a/b) and its complement · Area = a×b ÷ 2
Worked examples: legs 3 and 4 give c = √25 = 5 — the founding triple. Leg 6 with hypotenuse 10: √(100−36) = 8, completing 6-8-10 (the 3-4-5 doubled). Angles 36.9° and 53.1° in both, because scaling preserves shape.
Pythagorean Triples
Integer triangles — exact answers, no radicals — and their multiples:
| Triple | Family | Where it appears |
|---|---|---|
| 3-4-5 | ×2 → 6-8-10, ×3 → 9-12-15 | The builder's squaring check |
| 5-12-13 | ×2 → 10-24-26 | Textbook favorite #2 |
| 8-15-17 | — | The 'unexpected' exam triple |
| 7-24-25 | — | Near-degenerate and satisfying |
| 20-21-29 | — | Almost-isosceles rarity |
| 9-40-41 | — | Leg, leg+1 hypotenuse pattern |
Any triple scales into infinitely many (multiply all three sides) — and primitive triples all generate from Euclid's m,n formula, one of number theory's tidiest results.
The Theorem at Work
Construction's 3-4-5 check: measure 3 ft along one wall, 4 ft along the other — the diagonal reads exactly 5 ft only if the corner is square. Multiples (6-8-10, 9-12-15) stretch precision over longer runs. Stair stringers, rafter lengths, and fence diagonals are the same calculation with different names on the sides.
Screen sizes are hypotenuses: a '55-inch' TV is 55 diagonal — at 16:9 that's roughly 48 × 27 inches of actual screen, straight from the theorem. Ladders lean on it too (a 20 ft ladder with feet 5 ft out reaches √(400−25) ≈ 19.4 ft up the wall), and the 4-to-1 ladder rule is the theorem wearing a safety vest.