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Squares vs rectangles

September 18, 2017

One of the persistent misconceptions I come across as a teacher is the question of what exactly is it that makes a shape a rectangle? This is something I mentioned briefly on episode 48 of Wrong but Useful (coming soon here) but I think it’s worth getting to a wider audience. 

Chances are that if I ask you to imagine a rectangle, something like this will come to mind:

I suppose it’s fair to say that if you’re reading this then maybe it might be more like this one:

I think the reason the first rectangle I showed is most likely is because that’s how they usually appear in ‘my first shapes’ books. Note that they almost always have their edges parallel to the sides of the book and certainly have “two long sides and two short sides”. However the necessary requirements for a shape to be a rectangle are:

  • It’s a closed quadrilateral (a 2d shape with four straight sides and no gaps)
  • It has four right angles

That’s it. 


However, in our Equable Shapes project, students are asked to explore shapes where the perimeter and area have the same numerical value. We start with rectangles and sooner or later, someone finds that 4 by 4 works (area and perimeter both 16). At this point all hell breaks loose and the two sides are formed with the “you can’t have that because sir said it had to be a rectangle” gang squaring* off against the “squares are special rectangles” crew. 

Inevitably this leads to a lot of unpicking and most students believe they have been told explicitly at some point in the past that reactangles have two long sides and two short sides. Obviously I delicately correct this by going through what I mentioned above but some students still struggle to get that all squares are rectangles but it doesn’t follow that all rectangles are squares. The most effective way I’ve found to help illustrate the point is with furniture.

“Do you know what the word furniture means?”


“Are you happy that all wardrobes count as furniture?”


“Are you also happy that not all furniture is a wardrobe?”


“Then you can understand about the squares and rectangles.”

“Oh. I think I get it now.”

So, there we have it. Squares are rectangles, rectangles might be squares and oblongs are something for another post.

Why not try asking people you work with whether a square is a rectangle? The 

*pun very much intended

From → Maths

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