Return to my Puzzle pages
Go to my home page

Rectangle to Square Dissection

© Copyright 2003, Jim Loy

I received email about dissecting a rectangle to form a square (the email asked about dissecting into triangles, but that is a trivial variation of the general problem). Well, the most obvious approaches are frustrating, and I don't see the dissection at MathWorld. But the dissection is not particularly difficult. This is what I stumbled onto (I'm sure that it was known before I discovered it):

On the upper left, we have a rectangle, with sides a and b. We construct the shown length sqrt(ab), which is the geometric mean of a and b. See Squaring a Pentagon to see how to do that. We can then rearrange the three pieces shown to form the square to the right of the rectangle. The quadrilateral, inside the rectangle and square, can be dissected into two triangles, if triangles are what you need. Below that is the same dissection with a shorter, wider rectangle. A long thin rectangle must be dissected into a shorter, wider rectangle before using this method (rectangles shown on the right side of the diagram).

Any dissection in one direction can be done in the opposite direction. So, you can dissect a square into any rectangle of the same area.

Addendum:

A few rectangles can be dissected into a square with just two pieces. In this diagram, if the square is 1x1, then the rectangle is 4/5 x 5/4. With a different number of stairsteps, we can ge some other rectangle, like 1/2 x 2/1, or 2/3 x 3/2, or 3/4 x 4/3. This method does not work for the general (arbitrary) rectangle.

Return to my Puzzle pages
Go to my home page