How many rectangles of any size are on an 8x8 chess board?

Question

I know that the answer is 1296 (including squares) and that the equation is (1+2+3...8)(1+2+3...8)... but how does this formula work. Explain

Answer 1

You may think that the rectangles are constructed by sides along the x-axis and sides along the y-axis. Obviously, if there are N ways to pick different sides along the x-axis, there are also N ways to pick different sides along the y-axis. Therefore, we may restrict us on picking different sides along the x-axis.
There is only one longest side length, 8, from Grid 0 to Grid 8. However there are two next-short-side length 7: from Grid 0 to Grid 7, and from Grid 1 to Grid 8. Also, there are three next-short-side length 6, and so on. Therefore, there are (1+2+3+4+5+6+7+8) ways to pick the sides along the x-axis. , and there are (1+2+3+4+5+6+7+8)(1+2+3+4+5+6+7+8) rectangles of any size are on an 8x8 chess board.