# How do I cut a square in half?

I have a square that's $10m \times 10m$. I want to cut it in half so that I have a square with half the area. But if I cut it from top to bottom or left to right, I don't get a square, I get a rectangle!

I know the area of the small square is supposed to be $50$, so I can use my calculator to find out how long a side should be: it's $7.07106781$. But my teacher said I should be able to do this without a calculator. How am I supposed to get that number by hand?

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Actually, the exact side length is the square root of 50, which is irrational. –  Akhil Mathew Jul 20 '10 at 21:40

Does this give you any ideas?

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Wow, looking at the picture, I see that the square is evenly split into the black and white areas. I wasn't convinced at first, but then I printed out the picture and saw that the four black triangles can be 'folded' over to cover the small white square perfectly. –  Larry Wang Jul 20 '10 at 23:23
yup, each of the quadrants of the outer square is split in half. –  Jason S Jul 20 '10 at 23:24
...of course, if you're working in wood and the direction of the grain is important, this method would mess that up. –  Jason S Jul 20 '10 at 23:25
This is exactly the same figure that Plato used in his dialogue Meno (search for the first instance of "Boy", and read from there) to answer the very same question as the OP asked here. –  Per Manne Oct 17 '12 at 11:10

A simple approximation

x^2 = 50
7^2 = 49


so the number is around 7.

A even better approximation

(7+x)^2 = 50
14x + x^2 = 1
(14+x)x = 1
14+x = 1/x


This basically shows x is pretty small, and around 1/14, without a calculator, it's not likely you can find what exactly 1/14 is, but at least you know it be smaller than 1/10. 7.1 is a good approximation. (off by less than 0.03)

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My teacher said no estimating :( –  Larry Wang Jul 20 '10 at 21:46