You are using squared graph paper, right? When you have drawn a 2x2 square you just count how many squares of graph paper are enclosed. If the side of the square is 2 then the area enclosed is 4 squares.
But how do you draw a square with area 2? What you do is draw the diagonals of each of the 4 enclosed squares. For the top left and bottom right squares you draw the diagonal from the bottom left corner to the top right corner; and for the other two squares you draw the other diagonal. This gives you a square inside the bigger square tilted by 45 degrees.
You have also divided the area of 4 squares into 8 triangles. So 8 triangles have a total area of 4 little squares. Each triangle has an area of ½. The tilted square contains 4 of these triangles. So if 8 triangles is equivalent to 4, then 4 triangles is equivalent to 2. That's how you know the area of your tilted square is 2. So its sides have length √2, the length of each diagonal. No Pythagoras!
Looking at squares geometrically probably also helps you to answer your other question, when you start with the tilted square. Because the area of the tilted square is 2, its side length is √2. But the side is the diagonal of a unit square. This means to get the length of the side of the square knowing the length of its diagonal, you just divide the length of the diagonal by √2.
