The Math Less Traveled

An area paradox


Here’s a fun paradox which has been around for quite a while and was apparently a favorite of Lewis Carroll.

As you can verify for yourself, the two figures above are composed of two different rearrangements of the same four pieces. However, if we compute the areas of the two figures we get a surprise: the area of the square is , whereas the area of the rectangle is ! Where did the extra unit of area come from?

Feel free to post questions, comments, thoughts, guesses, etc. below. (However, please refrain from commenting if you have seen this paradox and its explanation before!)

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