The Pythagorean Theorem states that the square of the length of the
hypotenuse of a right triangle is equal to the sum of the squares of the lengths
of the legs. In other words, if c is the length of the hypotenuse, and the two
legs have length a and b, then

c² = a² + b²

A visual proof of this theorem follows:

Here you see three squares, with areas equal to the squares of the three
sides of the red triangle. The two smaller squares can be rearranged to
make the larger square, proving the sum of the two smaller squares is the larger
square.

Integer values of a, b, and c that make this equation true are called
Pythagorean Triples.

### Related Pages in this website

Go back to the Number Theory Home

Pythagorean Triples

Triangles

Formulas for Primitive Pythagorean Triples and
their deriviation -- a way to generate all the triples such that a^2 + b^2
= c^2

Prove that the area of a right
triangle with integer sides is not a perfect square. The proof is
here (with some help from someone from the nrich website)

Puzzle question -- if
m is the product of n distinct primes, how many different right triangles with
integer sides have a leg of length m?

Arithmetic Sequence of Perfect
Squares, page 3 -- If a^2, b^2, c^2 are in arithmetic sequence, why is
their constant difference a multiple of 24? Look at the second answer to
this question for the relationship between Pythagorean Triples and this
arithmetic sequence of squares.

Theorems Involving Perfect Squares
-- answers questions such as "why is the square root of x irrational unless x
is a perfect square?" and other fundamental questions about perfect squares.

The webmaster and author of this Math Help site is
Graeme McRae.