One way to figure it out is by reading Plato. In Meno, Socrates shows how you can construct a square with an area of eight feet from a square with an area of four feet (that is, with two-foot sides). To do this, join four squares with four-foot areas together to make a larger square with a sixteen-foot area. Then draw a line between the midpoints of the adjacent sides of the sixteen-foot square to make a new interior square (see diagram). This cuts each four-foot square in half along the diagonal. Since you have cut each four-foot square in half, you now have eight two-foot triangles, with four of the triangles joining together to make a square with an eight-foot area.
You can follow a similar procedure to create a square with a two-foot area from a square with one-foot sides. Since the area of a square is equal to the side length squared, the side length of a square with an area of two feet is the square root of the number two, now easily measurable.
Or you could skip the philosophy and just read Pythagoras instead! The square of the hypotenuse is equal to the sum of the squares of the other two sides.
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