Archimedes invented a division of a square into 14 pieces.  This preoccupation with division of squares lead to a game similar
to Tangrams involving making figures from the 14 pieces. The Chinese developed the game of Tangrams, a division of a square
into 7 pieces.  There are some interesting mathematical concepts involving tangrams.  It is also a lot of fun manipulating the
arrangement of the 7 pieces to form many creative shapes.  You may have seen tangram shapes in the book, Alice's
Adventures in Wonderland, written by Charles Dodgson, a mathematician and author writing under the pen name Lewis
Carroll.  (More info available on this topic at the Math Games History Page.)

The tangram pieces consist of 7 pieces:  2 large triangles, 1 medium triangle, 2 small triangles, 1 small square, and 1 small
parallelogram.

You can create tangrams by taking a square piece of construction paper, dividing the square into 16 smaller squares, and
cutting diagonal lines as shown below.

 

Assuming that the area of the entire square is 1, what is the area of each piece?

         Large triangle?   Medium triangle?
         Small triangle?   Small square?
         Small parallelogram?

         Add up the areas of all of the 7 pieces.  What is your answer?

 
Now assume that the area of the large triangle is 1, what is the area of each piece?

         Medium triangle?   Small triangle?
         Small square?   Small parallelogram?
         Entire square?

 
Now assume that the area of the small square is 1/5, what is the area of each piece?

              Large triangle?   Medium triangle?
              Small triangle?  Small parallelogram?
              Entire square?