Each triangle has an area of
Ö3
4

. Clearly if I can make a square out of such shapes then its area will be a whole number multiple of
Ö3
4



Now, the triangle has three lengths:

1, 0.5, Ö3
2

This means that the side of the square will be of length

N+ MÖ3
2
 + P
2

for some whole numbers N, M, P
Squaring this side length gives us a number of the form
A+BÖ3
where A and B are rational numbers with A not zero (B might be zero, but that does not concern us).

This contradicts the statement that the area of the square is a rational multiple of
Ö3 and, therefore, no such square is possible.