# Squares from diagonals

Take a look at the video showing squares and their diagonals...

*This resource is part of "Dotty Grids - Drawing Polygons"*

I wonder what maths we can do on dotty paper...

This video might give you some ideas.

Play with the dotted grid below, or print out some dotty paper.

Damini from Newstead Wood School For Girls discovered that not all diagonal lines that join two grid points make a square:

You find the centre of your diagonal line and go from there away from the line in a straight line (at right angles) until you find the right point. Then join the point to one end of the diagonal line and you have a side! How easy was that. Not all diagonal lines that join two grid points make a square. If you take a piece of square dotty paper and join up any two dots it will be hard to find the square, as there might not even be a square.

Megan and Kit from Tanglin Trust, Singapore, did some systematic experimenting and came up with a general rule.

They found that whether or not the diagonal makes a square depends on how many squares up and across you go.

As you can see from our table, each kind of diagonal only works (makes a square with corners on the dots) if you put two odds or two evens together. For example $3$ up by $3$ across makes a square but $3$ up and $4$ across won't.

Can you see why this is? Please send in any more thoughts here or on the page of the problem, email to

You find the centre of your diagonal line and go from there away from the line in a straight line (at right angles) until you find the right point. Then join the point to one end of the diagonal line and you have a side! How easy was that. Not all diagonal lines that join two grid points make a square. If you take a piece of square dotty paper and join up any two dots it will be hard to find the square, as there might not even be a square.

Megan and Kit from Tanglin Trust, Singapore, did some systematic experimenting and came up with a general rule.

They found that whether or not the diagonal makes a square depends on how many squares up and across you go.

Image

As you can see from our table, each kind of diagonal only works (makes a square with corners on the dots) if you put two odds or two evens together. For example $3$ up by $3$ across makes a square but $3$ up and $4$ across won't.

Can you see why this is? Please send in any more thoughts here or on the page of the problem, email to

*secondary.nrich@maths.org*For ideas on how this problem and others from the Dotty Grids Collections can be used in the classroom, you may be interested to read this article.

A printable version of this problem is available as a Word or Pdf file.