Pythagorean Triples

A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule:

a2 + b2 = c2

Example: The smallest Pythagorean Triple is 3, 4 and 5.

Let's check it:

32 + 42 = 52

Calculating this becomes:

9 + 16 = 25

And that is true

Triangles

Pythagoras

A triangle with a Pythagorean Triple has a right angled triangle (see Pythagoras' Theorem for more details):

pythagoras squares a^2 + b^2 = c^2

Note:

  • c is the longest side of the triangle, called the "hypotenuse"
  • a and b are the other two sides

 

Example: The Pythagorean Triple of 3, 4 and 5 makes a Right Angled Triangle:

triangle 3 4 5

triangle 3 4 5 lego

Here are some more examples:

5, 12, 13   9, 40, 41
5,12,13 Triangle   9,40,41 Triangle
52 + 122 = 132   92 + 402 = 412
25 + 144 = 169   (try it yourself)

 

And each triangle has a right angle!

List of the First Few

Here is a list of the first few Pythagorean Triples (not including "scaled up" versions mentioned below):

(3,4,5) (5,12,13) (7,24,25) (8,15,17) (9,40,41)
(11,60,61) (12,35,37) (13,84,85) (15,112,113) (16,63,65)
(17,144,145) (19,180,181) (20,21,29) (20,99,101) (21,220,221)
(23,264,265) (24,143,145) (25,312,313) (27,364,365) (28,45,53)
(28,195,197) (29,420,421) (31,480,481) (32,255,257) (33,56,65)
(33,544,545) (35,612,613) (36,77,85) (36,323,325) (37,684,685)
... infinitely many more ...

Scale Them Up

The simplest way to create further Pythagorean Triples is to scale up a set of triples.

Example: scale 3,4,5 by 2 gives 6,8,10

Which also fits the formula a2 + b2 = c2:

62 + 82 = 102

36 + 64 = 100

If you want to know more about them read Pythagorean Triples - Advanced

 

 
Pythagoras Theorem Right Angled triangles Pythagorean Triples - Advanced Triangles Proof that a Triangle has 180°