The square on the left is made up of two identical triangles and two identical trapeziums. Each triangle's base is 16 cm and height 6 cm. Each trapezium's height is 10 cm. The length of the parallel sides of each trapezium is 6 cm and 10 cm.

The four pieces are repositioned as in the figure on the right. The area of the figure on the left is 256 cm² (16 cm x 16 cm) but the area of the figure on the right is 260 cm² (10cm x 26 cm).

Why is the area increased by 4 cm² after the pieces are repositioned?

This puzzle is similar to quiz #6. Careful examination of the figure on the right will convince you that it is not a rectangle and its area, therefore, is not 10 cm x 26 cm.

If the figure was a rectangle, the triangle formed by the diagonal (area I and III) would be congruent to the triangle denoted by area I. The ratio of the two vertical sides of triangle I is 6/16 or 3/8 while the two vertical sides of the area formed by area I and III has a ratio of 10/26 or 5/13. Since it is given that area I is a triangle, the area formed by area I and III cannot be a triangle and the whole figure is not a rectangle.