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Tilted Squares

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


Age 11 to 14 Challenge Level:

We build an imaginary tower of squares inside a right angled isosceles triangle. The largest square stands on the hypotenuse of the right angled triangle. Each square has two vertices touching the other sides of the triangle. Only three squares are drawn in the diagram but imagine that there are infinitely many getting smaller and smaller and smaller...

What fraction of the area of the triangle is covered by the squares?

You can do this without a lot of calculation and without any advanced mathematics. If you wish to extend this project you can ask: What if the triangle was equilateral? Or what if the tower was made up of rectangles? Or...