# Beyond Pythagoras

William Kerslake        Mathematics GCSE Coursework

Beyond Pythagoras

## Aim

I am conducting this investigation to discover formulae that will allow me to calculate many Pythagorean triples. I will first find formulae for odd Pythagorean triples and then even ones. For each, I will find formulae for ‘a’, ‘b’, ‘c’, perimeter and area, all in terms of ‘n’. After this I hope to discover a general formula for all Pythagorean triples, although it is unlikely that I will.

Contents

Odd Pythagorean Triples

Table of first 10 odd triples

Even Pythagorean Triples

## Table of first 10 even triples

To work out the area and perimeter, I used the known formulae;

Perimeter = a + b + c

Area = a x b

2

## Odd Triples

First, I need to work out ‘a’ in terms of ‘n’:

The differences between the numbers show that ‘a’ is changing twice as fast as ‘n’. Therefore, 2n must be part of the formula.

The differences show that ‘a’ and ‘n’ are changing at the same rate – increasing by 2 each time, thus proving that the formula includes 2n. The difference between ‘a’ and ‘n’ is always 1; therefore, our formula needs to have a +1 included.

From this, I know the formula for ‘a’, in terms of ‘n’, to be;

a  =  2n + 1

I personally cannot see an obvious link between ‘n’ and ‘b’ or ‘c’, ...