First, I tried this by drawing a rectangle on a sheet of graph paper that was say, 40 units in area. Then, I would try to draw a different shape that was 40 units in area. This was too difficult though, because with basically all other shapes besides squares/rectangles, not all of the individual units would be completely filled, so it was hard to calculate exactly how many units the new shape took up.
I found that it was easiest to take a square, for example, and then cut it up into smaller triangles, squares, etc., and then use those smaller shapes to construct a new shape with the same area as the previous one.
I started with a rectangle (50 units in area):
Then, I cut it up into lots of smaller triangles, and one one by ten unit long rectangle. I put them back together, and ended up with something like this:
I did the same for an octagon, but this time, I am posing it as a challenge for you guys. Here is the shape I got upon dividing an octagon, and then re-constructing it (using the method above). How could you use this to make an octagon? what would the area be? Let me know how you approached this problem.