What is the Pythagorean theorem?


The Pythagorean theorem (Pythagoras' theorem) states that in a right (angled) triangle the sum of the squares of the legs a and b equals the square of the hypotenuse c

a²   +   b²   =   c²



Pythagorean triples


A Pythagorean triple is a set of positive integers which fits Pythagoras' theorem.


The simplest example of a Pythagorean triple is a = 3, b = 4 and c = 5 units

a²   +   b²   =   c²

3²   +   4²   =   5²


Other examples include a = 6, b = 8, c = 10 units

6²   +   8²   =   10²


and a = 12, b = 16, c = 20 units

12²   +   16²   =   20²


Notice how a, b and c are always divisible (capable of being divided) by 3, 4 and 5 respectively.


But a, b and c don't have to be whole numbers!


For instance, if a = 1.5 and b = 2 units

a²   +   b²   =   c²

1.5²   +   2²   =   c²

2.25   +   4   =   c²

   6.25   =   c²

6.25   =   c

2.5   =   c

c   =  2.5 units


To square a number on your calculator you'll need to use x². You can do the above calculation by entering 1.5, x², add sign (+), 2, x², equals sign (=), square root sign () - you may need to press a shift or 2nd function button to access x²

What if a = 0.75 and c = 1.25 units?

a²   +   b²   =   c²

0.75²   +   b²   =   1.25²

0.5625   +   b²   =   1.5625

 b²   =   1.5625   -   0.5625

b²   =   1

b²   =   1

b   =   1 unit


Grab your calculator again and practice using the x² and buttons - you should get the same values as in the above calculation.


And how about if b = 6 and c = 7.5 units?

a²   +   b²   =   c²

a²   +   6²   =   7.5²

a²   +   36   =   56.25

 a²   =   56.25   -   36

a²   =   20.25

a²   =   20.25

a   =   4.5 units


Pythagorean theorem proof


How can you prove Pythagoras' theorem?


Imagine a square which contains

  • a smaller square with side c
  • 4 identical Pythagorean triangles, each with legs a and b and hypotenuse c

 


Now you can re-arrange the triangles in a such a way that you end up with two even smaller squares, one with side a, the other with side b



Note how the overall dimensions remain the same, so the area shaded a tan color in the first diagram (i.e. c²) must be equal to the area shaded tan in the second diagram (i.e. a² + b²).


So c² = a² + b² which is the same as

a²   +   b²   =   c²


Let's summarize!


Remember the Pythagorean theorem relates to right (angled) triangles only.


If you know the length of two of the sides a, b or c then you can work out the length of the third side using the formula

a²   +   b²   =   c²


› Pythagorean theorem