Suppose we are given two polygons P and Q with equal area. Will it always be possible to divide one of them using straight cuts into pieces and rearrange them to exactly cover the other? Try it for:

(A) a square and a non-square rectangle of equal area,
(B) two differently shaped triangles of equal area,
(C) a triangle and a square of equal area.
Formulate a conjecture of your own about this.                                                                                                        [Page No. 134]

In every case the answer turns out to be yes, and the three examples point to a general rule.

(A) Square and non-square rectangle: A rectangle can be cut into a small number of pieces (using a 'staircase' or slanting cut) and rearranged into a square of the same area, so these two are dissectable into each other.

(B) Two differently shaped triangles of equal area: Any triangle can first be turned into a rectangle of equal area: cut along the line joining the midpoints of two sides (parallel to the base) and fold the top piece down. Each triangle becomes a rectangle, and any two rectangles of equal area can be cut into each other and the two triangles can too.

(C) A triangle and a square of equal area: Convert the triangle to a rectangle of equal area as above, then convert that rectangle to a square (Baudhāyana's rectangle-squaring construction). So a triangle and a square of equal area are dissectable into each other.

Conjecture: Any two polygons with the same area can be divided by straight cuts into a finite number of pieces and rearranged to exactly cover one another. The trick that makes it work is to turn every polygon into a rectangle (via triangles), since any two rectangles of equal area can be cut into each other.