An Appearing Act

White rabbit on grass

Draw an eight by eight square on dotty paper. You can print some here.


Cut it up to make the four pieces shown in the diagram:

eight by eight square, divided into two trapezia and two triangles
Now rearrange the pieces to make the shape below:
pieces of square rearranged to form 13 by 5 rectangle

It seems that the same pieces can make two shapes with different areas!

Can you explain where the extra area comes from?


Here are some questions you might like to consider:

Can other squares be split up and rearranged to make rectangles with a different area?

Are there other square/rectangle pairs where the areas differ by 1 square unit?

Is there a pattern in the sizes of squares that can be arranged in this way?





Yes, but can you explain it?