Is a square a rectangle?

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I like this question. My first reaction — since I get this pretty much every semester that I teach a problem-solving or geometry class — is to ask what the definition of a rectangle is. Most people respond that it’s a quadrilateral with 4 right angles, maybe they add something about the opposite sides being parallel and/or equal, and then I ask if a square fits that definition. They answer yes, and the problem is solved.

But I think the question is really a little more subtle than that. In all the children’s books that we’ve acquired on shapes, none of them show a square on the rectangle page. Years of reinforcement that squares and rectangles are different shapes is hard to overcome with a single definition.

Furthermore, when I started teaching Geometry I learned that 2300 years ago Euclid didn’t define rectangles (which he called oblongs) in quite the same way as we do. Here’s a page from Oliver Byrne’s 1847 translation of Euclid’s Elements, which is one of my favorites because Byrne sure liked his color markers. He uses oblong the way we use rectangle.

Notice that Euclid said that an oblong did not have all four fides equal: a fquare was a completely different beast, not a special kind of rectangle. Euclid kept this distinction with all his geometric figures: a rhombus couldn’t have right angles (so a square wasn’t a special kind of rhombus either), a parallelogram (rhomboid) did not have right angles or equal sides, and an isosceles triangle had exactly two equal sides, not at least two. At Euclid’s Geometric Figures party when the figures divide into teams, the squares knew EXACTLY where to go, and it wasn’t with the rectangles: it was a partition, rather than a Venn diagram.

Another place where geometric problems can occur is with triangles. I think of the stereotypical triangle [in the US — is it true in other countries as well?] as being one with a horizontal base, and probably isosceles.

But, just like the definition of rectangle, that hasn’t always been the case. In in “Words and Pictures: New Light on Plimpton 322”, Eleanor Robson explains, “if we look at triangles drawn on ancient cuneiform tablets like Plimpton 322, we see that they all point right and are much longer than they are tall: very like a cuneiform wedge in fact.”

Neither triangle is better or worse than the other, but they are different, illustrating the cultural influence on mental images of shapes. I find that intriguing.

I believe that the page of Byrne’s translation is fair to include because its over 70 years old. And an edition only sold for $300 in the ’70s — can you believe it? Not that I had more than $5 at any one time in that decade, but still, if I had and I wasn’t buying dollhouse furniture, I’m sure I would have bought it.

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11 Responses to “Is a square a rectangle?”

  1. educatorblog Says:

    Wow, intriguing post. I think that part of visual literacy is being able to understand how culture and history impact how we view objects. (educatorblog.wordpress.com)

  2. Yet Another Heptagon « 360 Says:

    […] Not on the heptagon page: I was actually looking up copyright info for the blue triangle I used in this post, and I noticed that the copyright statement, explaining that the figure was a simple geometric […]

  3. Is a Square a Rectangle?: Welsh edition « 360 Says:

    […] posted before about how even though in the US we define squares to be equilateral rectangles, there are many […]

  4. CArmel Says:

    I don’t like your answer

  5. Russel Says:

    the square is not like rectangle because the all sides of square are all equal and all angles are right angles while the rectangle we have four right angles and opposite sides are equal and parallel and the sides of rectangle is not all equal.

  6. Ξ Says:

    Carmel, the question usually comes because students remember something about a square being a rectangle, or a rectangle being a square, but they aren’t sure which is which. It seems to work well enough in the classes where it comes up, but I can absolutely imagine that there are other good answers to the question.

    Russel, I’m not sure I understand your point. In the United States, the definition of rectangle that appears in high school geometry texts (or Wikipedia) doesn’t say anything about whether or not the sides are equal. This means that rectangles can have longer and shorter sides, but it’s also allowable for the four sides to be the same length. In that sense, a square is just a special kind of rectangle, the way an equilateral triangle is a special kind of triangle.

    I believe this definition does vary by country, however, and so in countries that define rectangles as having sides that are not all equal, a square would not be a rectangle.

  7. Scott Says:

    I just had a disagreement with my son’s elementary school teacher about this. A square should not be considered a rectangle. Look at the larger sets – let’s start with “simple polygons” then the subset “quadrilaterals / quadrangles” then the subset of those called “parallelograms” In here we have a subset called “rhombus”. A square is a “rhombus” with a right angle (4 right angles in fact). A rectangle is a “parallelogram” with a right angle that is NOT a rhombus. Since rectangles don’t fit into the rhombus subset, one should not call a square a “special type of rectangle”. A rectangle with all 4 sides of equal length is a square, period. Should we teach our children to call squares rectangles? I think not. The very fact that the school wants to, and in fact makes sure that they consider squares to be rectangles is pointless. It is generally accepted that we call something by it’s most specific trait. Thereby, we glean the most information from that definition. Should we do away with boy, girl, man, woman, and just say “I see four humans”? Or, should we say “I see 10 automobiles”, rather than I see 4 cars, 2 minvans, 1 SUV, and 3 pickup trucks”? No, we lose information that way. Maybe we should teach the kids to call all shapes with sides polygons and be done with it. There are people in the world who want to consider Pluto to not be a planet anymore, so why can’t the mathematics community get together and kick out the “a square is a rectangle” argument. It makes more sense to consider them differently if you look at the larger sets and their subsets. A square is a rhombus, it is a parallelogram, it is a quadrilateral, it is a quadrangle, it is a simple polygon — and some say it is a rectangle. All these make sense, logically, except to call a square a rectangle. They are given two different names to distinguish them, so why can’t we just do that – distinguish between the two instead of getting hung up on the “a square is a special rectangle” portion? It really isn’t — a squre is a square, and a rectangle is a rectangle — or maybe we should do away with any distinguishing features so as not to offend any shapes and just call every shape with sides a polygon. Is it really that much of an intellectual leap to call a square a rectangle? I think that it is a much larger intellectual leap to see how rectangles don’t fit into the rhombus subset, and to then conclude that a square, therefore, IS NOT A RECTANGLE.

  8. Batman Says:

    “They are given two different names to distinguish them”

    Here’s where I have to disagree with Scott, at least as a mathematician. In mathematics, it is generally a Good Thing if we can show that two objects with different names are actually the same kind of object. Doing so allows us to apply results about one to the other. For example, a vector space is really just a special kind of module over a field (which is itself a special kind of ring). So whenever we learn something about modules, we can say the same thing about vector spaces – we gain information this way.

  9. roberto Says:

    “If you have a wire 38 cm long, what is the largest rectangle you can enclose with it?” Calculus has something to say.

  10. Paul Says:

    Rectangles are all shapes with a right-angle in each corner and two pairs of parallel sides. A square is a subset of this group/special kind of rectangle where all sides are also equal. The subset of rectangles excluding squares are called oblongs.

  11. Bob Saget Says:

    This was a highly educative anecdote, I must agree. I particularly liked the way rhombuses were mentioned as they don’t get as much credit as squares do. In actuality, the rhombus is a patriotic symbol as well as a symbolic one. God bless.

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