9.2 Halving a Square

The opposite task is equally handy: starting from a square, produce one with half the area. Simply reverse the earlier construction, sketch a smaller tilted square inside the original by joining the midpoints of its sides.

half area

Would halving the side length halve the area? Not at all. A side of half the length gives area \tfrac12 \times \tfrac12 = \tfrac14 of the original, it would take four such tiny squares to refill the original, not two. The half-area square is the tilted inner square on the midpoints, which divides the original into pieces that pair up two-into-one.

Try This: Halving with Paper Take a square sheet. Fold each corner inwards so the creases run through the midpoints of the sides. The inner diamond PQRS left behind is a square whose area is half the original. Draw the diagonals QS and PR, examine the four right triangles that appear, and use triangle congruence to satisfy yourself that PQRS is genuinely a square and genuinely half the area.