The table below shows constructions for squares with areas 1 through 64 using the following four methods and their combinations:

a) Squares with whole number roots (as 1, 4, 9).

b) Squares that are half (or double) the areas of other squares (as 2, 8).

c) Squares whose root is the diagonal of a rectangle (as 5, 10).

d) Squares whose root is the height of a triangle (as 3).

Squares with the area 0f 6 and 7 are combinations of methods b) and d):

6 is half of a square with the area of 12, whose root is the height of a triangle with base = 2, hypotenuse = 4.

7 is (intended to be) quarter of a square with the area of 28, whose root is the height of a triangle with base = 6, hypotenuse = 8

*There is an error in this drawing whereby the square that was intended to have an area of 7 instead has an area of 9 3/4. This is because I started with a rectangle of 8x5 rather than 8x6 (8*8 - 6*6 = 28 * 1/2 = 14 * 12 = 7 whereas 8*8 - 5*5 = 39 *1/2 = 19 1/2 * 1/2 = 9 3/4)! The principles are all good though!*