Some polygons do not tessellate easily. An example is the octagon:
Overlaying this with a tessellation of squares gives eight pieces. But instead the octagon tessellates very easily if we combine it with a small square:
We can form another tessellation by combining the same small square with a square of the same area as the octagon:
Since these two tessellations repeat in the same way we can overlay them to form the following dissection:
What we have done is to create a dissection of a square with a small square to an octagon with the same small square. But we can throw away the small squares giving a simple dissection of a square to an octagon.
This requires only five pieces, which is a very good improvement.
This method is not often used, but when it is, it gives very nice dissections. Some examples are the dissections of hexagon to dodecagon, octadecagon or hexagram. Another is the dissection of a square to latin cross.