A circular disk can be cut into two semicircular pieces easily by cutting along a diameter.

Can the disk be cut into pieces which can be arranged into two semicircles, with none of the cuts passing through the center?

                 *A*
        *         |          *
      M--J--S--X--U-----N
     *    |   |        |       *
     *    |   |  C    |       *
     *    |   |        |       *
      O--K--T--Y--V-----P
        *          |         *
                  *B*


AB is a diagonal which is intersected by a pair of parallel lines, MN and OP at X and Y, and are equidistant from centre C.

ST and UV are a pair of parallel lines equidistant from and parallel to AB.

JK is parallel to ST with JS = SX.

Cut along all lines and on reassembly J occupies the position of X and X occupies J's original position.

 The construction creates a rectangle whose area is the sum of two smaller rectangles having areas in the ratio of 1:2.
    

Edited on April 27, 2007, 10:25 pm

Posted by brianjn on 2007-04-27 22:22:11