Beyond Pythagoras

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William Kerslake        Mathematics GCSE Coursework

Beyond Pythagoras


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.


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


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.

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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’, ...

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