Tilted squares

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Problem

Tilted Squares printable sheet



It's easy to work out the areas of squares drawn on a grid if they are oriented in the usual way:

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These squares have their bases horizontal.



 

Can you find a quick and easy method to work out the areas of tilted squares?

 

Here are some squares with a tilt of 1:

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In these squares, the left bottom corner is one row lower in the grid than the right bottom corner.
See Getting started for suggested ways to calculate their areas.



Notice anything special about their areas?

Can you predict the areas of other squares with a tilt of 1?

 

What about squares with a tilt of 2? Or 3? Or 4? Or...?

Notice anything interesting?

Can you make any conjectures about the areas of tilted squares?

Can you prove your conjectures?

You might like to use the interactivity below to help you to draw tilted squares.

Full Screen tablet/mobile friendly version