## How to turn a rectangle into a square of the same area. |

Many thanks to MICHAEL JONES for supplying a neat little proof as to why this works!

Call the long side of the rectangle **a** and the short side **b**. The sides of the square are **s**. For the areas of the rectangle and square to be the same then **ab** must equal **s ^{2}**.

If you make a little right angled triangle as shown, the hypotenuse (the longest side) = **(a+b)/2** and one of the short sides is **(a-b)/2**. The other short side is **s**.

Pythagoras Theorem says that **s ^{2} = [(a+b)/2]^{2} - [(a-b)/2]^{2} **

If we bang this through...

**s ^{2} = (a^{2} +2ab + b^{2})/4 - (a^{2} -2ab + b^{2})/4
**

**s ^{2} = (a^{2} +2ab + b^{2} - a^{2} + 2ab - b^{2})/4
**

**s ^{2} = 4ab/4
**

**s ^{2} = ab **

...and that's what we wanted!

The Murderous Maths of Everything