# Definition:Pythagorean Triangle

## Definition

A **Pythagorean triangle** is a right triangle whose sides all have lengths which are integers.

## Examples

### $3-4-5$ Triangle

The triangle whose sides are of length $3$, $4$ and $5$ is a primitive Pythagorean triangle.

### $6-8-10$ Triangle

The triangle whose sides are of length $6$, $8$ and $10$ is a Pythagorean triangle.

This is not a primitive Pythagorean triangle.

### $5-12-13$ Triangle

The triangle whose sides are of length $5$, $12$ and $13$ is a primitive Pythagorean triangle.

### $7-24-25$ Triangle

The triangle whose sides are of length $7$, $24$ and $25$ is a primitive Pythagorean triangle.

### $693-1924-2045$ Triangle

The triangle whose sides are of length $693$, $1924$ and $2045$ is a primitive Pythagorean triangle.

### $4485-5852-7373$ Triangle

The triangle whose sides are of length $4485$, $5852$ and $7373$ is a primitive Pythagorean triangle.

### $3059-8580-9109$ Triangle

The triangle whose sides are of length $3059$, $8580$ and $9109$ is a primitive Pythagorean triangle.

### $1380-19 \, 019-19 \, 069$ Triangle

The triangle whose sides are of length $1380$, $19 \, 019$ and $19 \, 069$ is a primitive Pythagorean triangle.

## Also see

## Source of Name

This entry was named for Pythagoras of Samos.

## Historical Note

**Pythagorean triangles** were known to the ancient Babylonians in about $2000$ BCE.

The cuneiform tablet *Plimpton $\mathit { 322 }$* list $15$ sets of Pythagorean triples.

It is also clear that the author of that tablet was also familiar with the fact, proved in Solutions of Pythagorean Equation, that the numbers $2 p q$, $p^2 - q^2$ and $p^2 + q^2$ always make a Pythagorean triangle.

It is likely that the ancient Greeks, and in particular Pythagoras or one of his disciples, obtained this result from the Babylonians.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$