How many different ways can you divide a square into 4 parts of equal area?
Answer
Infinitely many. Start with any one division into 4 equal parts — say the two lines through the centre that cut the square into 4 small squares — and then reshape the pieces without changing their areas.
Take the boundary between two neighbouring parts and push it into one part. That part loses some area and its neighbour gains exactly the same area. So push the boundary in at one place and push it out by the same amount somewhere else along the same boundary:
Every bump pushed out of one part is a dent of the same size pushed into its neighbour, so all four areas stay equal.
Why it happens: the four parts together always make up the whole square. If a piece of area x is moved from one part to the next, and an equal piece of area x is moved back, both parts end with the area they began with. Since the size of the bump can be chosen in infinitely many ways, there are infinitely many such divisions.
Q2.
Try to think of different creative ways to divide a square into 4 parts of equal area. [Math Talk]
Answer
Here are four genuinely different families to try:
Four strips. Cut the square into 4 equal strips with three parallel lines. Each strip is side × (side ÷ 4) = one quarter of the square.
Four triangles from the centre. Join the centre to the four corners. Each triangle has base = side and height = half the side, so its area is ½ × s × (s/2) = s²/4.
Any four lines through the centre, at 45° apart? Not quite — but any two perpendicular lines through the centre do work, whatever their slope. The two lines cut the square into 4 pieces that map onto each other under a quarter-turn about the centre, so they are congruent.
Bumpy pieces. Start with any of the above and trade equal bumps and dents across a boundary, as in Q1.
Check it yourself: for the four triangles from the centre, add the four areas: 4 × s²/4 = s². They do fill the square exactly.