College of Arts and Sciences

Department of Mathematics

This is sometimes called the proof without words. The dimension of both squares is a+b. Since the same four triangles are contained in both squares, the area "left over" from the squares must be equal in both squares. It is easily seen that in the figure on the left, this area is c² whereas in the figure on the right, it is a²+b². This proof can be very effective if the square is drawn on an overhead transperancy, and the four triangles are cut from paper. After projecting the figure on the left, the upper left triangle can be rotated down to form a rectangle with the lower left triangle, and the triangle in the upper right can be rotated down to form a rectangle with the triangle in the lower right. The lower right rectangle can be slid up to the top of the square to give the figure on the right.