# A circle cut into smaller pieces can become a smaller circle?

While I was on a plane, a Math professor once told me that it was possible to divide a circle in multiple smaller pieces in such a way that, when those smaller pieces are assembled in another way, they create another circle, but this circle somehow is smaller than the first one.

-
Not true. But true of a $3$-dimensional ball, if you interpret "pieces" as subsets. See the Banach-Tarski paradox (Wikipedia). –  André Nicolas Nov 28 '11 at 20:38
It sounds similar to the Banach-Tarski paradox (en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox). I do not think this is possible in two dimensions using finitely many pieces; see terrytao.wordpress.com/2009/01/08/… . –  Qiaochu Yuan Nov 28 '11 at 20:38
Banach-Tarski is what you want, although the word "cut" is a misnomer, because the division of the points on the sphere doesn't match any intuitive notion of "cut." –  Thomas Andrews Nov 28 '11 at 20:43
It could very well be that he was talking about a ball instead of a circle, and that I misremembered. Thanks very much, all three of you! :) –  Qqwy Nov 28 '11 at 20:45

However, there is a famous related result, called the Banach-Tarski paradox, that says that the $3$-dimensional ball of radius $1$ can be decomposed into a finite number of sets, which can then be reassembled to make two complete $3$-dimensional balls of radius $1$. There is a large family of related results. The one that is closest to your question is that for any $r$ and $R$, a ball of radius $r$ can be decomposed into a finite number of sets which can be reassembled to make a ball of radius $R$.
Unrelated to your question, but interesting, is the following beautiful theorem of Laczkovich. The disk of area $1$ can be decomposed into a finite number of sets which can be reassembled to make a square of area $1$.