Pythagorean Theorem Calculator
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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.
Frequently Asked Questions
How do I find the hypotenuse?
Square both legs, add, square-root: legs 3 and 4 give √(9+16) = 5. The hypotenuse is always the longest side, opposite the right angle — if your 'hypotenuse' comes out shorter than a leg, the inputs are swapped.
How do I find a missing leg?
Rearrange: leg = √(hypotenuse² − other leg²). Hypotenuse 10, leg 6: √(100−36) = 8. Enter leg first, then hypotenuse in leg mode — the calculator guards the ordering because c² − b² must stay positive.
What is the 3-4-5 rule?
The smallest integer right triangle, used as a squaring tool: mark 3 and 4 units along two edges — a 5-unit diagonal certifies the corner square. Builders scale it (6-8-10 over longer runs) for the same check with less relative error.
Does the theorem work on non-right triangles?
No — it characterizes right angles exactly. General triangles upgrade to the law of cosines: c² = a² + b² − 2ab·cos(C), which collapses to Pythagoras when C = 90° (cos 90° = 0). The comparison a²+b² vs c² even diagnoses the angle: greater = acute, less = obtuse.
How does a TV's diagonal relate to its width?
Via the theorem and the aspect ratio: at 16:9, width = diagonal × 0.872 and height = diagonal × 0.490. A 65-inch set is about 56.7 × 31.9 inches — the wall-space number the marketing diagonal politely omits.
What made this theorem so famous?
It's the bridge between algebra and geometry — distance itself is defined by it (the distance formula is Pythagoras in coordinates), and it generalizes everywhere: 3D diagonals, vector magnitudes, even the complex plane. More proofs exist for it than any theorem in mathematics — several hundred, including one by a US president.
Methodology. This calculator uses standard, peer-reviewed mathematical formulas. It is reviewed and maintained by the Vast Calculators editorial team.
Last updated · July 11, 2026
Results are estimates for general use; verify critical figures independently.