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Squaring the Rectangle

Age 14 to 18 Challenge Level:

Why do this problem?

This problem encourages students to think deeply about area and length in order to prove a theorem. The suggestions and interactivity in the task provide some scaffolding to help students to investigate, make conjectures, and hopefully prove some of their conclusions. To get to a complete proof will require some perseverance.

Possible approach

You may wish to spend some time working on Triangle Transformations before starting on this problem.

"Draw a rectangle with any dimensions that you like. Your challenge is to find a way to cut it up and reassemble it to make a square."

Students may start by picking particular rectangles that can be easily rearranged. After they have had some time to explore, you may wish to share the interactivity below:

Invite students to work out how the square has been cut up, and how they could reverse-engineer this to start with a rectangle and finish up with a square. 

Alternatively, you might wish to share the image below, and invite them to consider how the pieces could be rearranged to make a square.

It might be fruitful to discuss the side length of a square which has the same area as a rectangle with sides of length $a$ and $b$: "If the top right corner of the rectangle has coordinates (a, b), how could you calculate the coordinates of the other dots?" 

Key questions

How do the pieces move to turn a square into a rectangle and back again?
How could you identify the points which are needed to make the pieces?
Can every rectangle be cut up and reassembled to make a square?

Possible extension

Students could consider other polygons and how to dissect them and reassemble them to make squares. They could find out about the Wallace-Bolyai-Gerwein Theorem, and perhaps the analogous problem in three dimensions, which is Hilbert's Third Problem.

Possible support

Students could use squared paper and construct squares, and then find rectangles with the same area - for example, turning a $6 \times 6$ square into a $9 \times 4$ rectangle, using the dissection shown above.