You may also like

problem icon

Rationals Between...

What fractions can you find between the square roots of 65 and 67?

problem icon

Equal Equilateral Triangles

Using the interactivity, can you make a regular hexagon from yellow triangles the same size as a regular hexagon made from green triangles ?

problem icon

An Introduction to Irrational Numbers

Tim Rowland introduces irrational numbers

The Square Hole

Age 14 to 16 Challenge Level:

Incidentally, did you notice that the yellow and purple triangles have the same area ? This doesn't require the particular case of one triangle being equilateral, any rectangle split into 4 areas by its diagonals will do.

More obvious now ?

Anyway back to the area of the Square Hole :

Thank-you to Clem, and to Marta & Brittany from MaST Community Charter School, and others who sent in solutions.

Seeing the image as a 'hole' surrounded by four rectangles, with each rectangle made from $2$ yellow (equilateral) and $2$ purple triangles.

The 'height' of the equilateral triangles is $\sqrt{3}$ divided by 2

So the dimensions of each rectangle are $1$ and $\sqrt{3}$

The side of the square hole is therefore $\sqrt{3} - 1$