📐 Pythagorean Theorem Calculator

a² + b² = c²

Side a (leg)

²+

Side b (leg)

²=

Side c (hypotenuse)

Try:

Hypotenuse c

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🔢 Pythagorean Triples

Pythagorean triples are sets of three integers (a, b, c) where a² + b² = c². Click any triple to calculate it.

🔬 Generate Triples

Using Euclid's formula: choose m > n > 0 to generate a primitive triple.

m (> n)

n (< m)

🌍 Real-World Applications

📐 Pythagorean Theorem — Full Guide

The Theorem

In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides.

a² + b² = c²

Where: a, b = legs (shorter sides)
c = hypotenuse (longest side, opposite 90°)

Finding c: c = √(a² + b²)
Finding a: a = √(c² - b²)
Finding b: b = √(c² - a²)

Euclid's Formula for Pythagorean Triples

For any m > n > 0, this generates a Pythagorean triple:

a = m² - n²
b = 2mn
c = m² + n²

Example: m=2, n=1
a = 4-1 = 3, b = 4, c = 4+1 = 5
→ (3,4,5) ✓ → 9+16=25

Distance Formula (Pythagorean Theorem in 2D)

Distance = √((x₂-x₁)² + (y₂-y₁)²)

This IS the Pythagorean theorem! The horizontal
difference is leg a, vertical difference is leg b,
and the straight-line distance is hypotenuse c.

Example: From (1,2) to (4,6):
= √((4-1)² + (6-2)²) = √(9+16) = √25 = 5

3D Pythagorean Theorem

For a box with dimensions l, w, h:
Diagonal = √(l² + w² + h²)

Example: 3×4×5 box:
Diagonal = √(9+16+25) = √50 = 5√2 ≈ 7.07

Right Triangle Trigonometry

sin(A) = a/c = opposite/hypotenuse
cos(A) = b/c = adjacent/hypotenuse
tan(A) = a/b = opposite/adjacent

Pythagorean identity: sin²(A) + cos²(A) = 1
(This is just a²/c² + b²/c² = (a²+b²)/c² = c²/c² = 1)

Who Was Pythagoras?

Pythagoras of Samos (570-495 BCE) was a Greek
philosopher and mathematician. However, the theorem
was known to Babylonians ~1000 years earlier!

The Plimpton 322 clay tablet (1800 BCE) lists
15 Pythagorean triples, including 13320-13500-18541.

Egyptian rope-stretchers used the 3-4-5 triple
to create right angles for pyramid construction.

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📐 Pythagorean Theorem Calculator