We get many questions about classifying shapes, from both elementary and high school students (or their parents or teachers). They often have trouble with the very idea of classifying items by applying definitions, and also with the fact that definitions can vary, both between everyday and technical usages, and from one textbook to another.
Is an X a Y? Can an X be a Y?
We see this question in several forms:
- Is a square a rectangle?
- Can a rectangle be a square?
- Are all squares rectangles?
I find that many students struggle with the very form of the question, as much as with the definitions it is asking about: Are we being asked if every square is a rectangle, or just if some particular square is a rectangle? A big part of the difficulty is that kids are not used to technical language, where the details of the wording are essential.They may need to be shown that, for example, “Is an X a Y?” means, “Is every X a Y?” (or, “Is an X always a Y?”), and not “Is some X a Y?”. On the other hand, “Can an X be a Y?” means, “Is some X a Y?” (or, “Are there any X’s that are Y’s?”), and not “Is it possible that every X is a Y?”. At a much more advanced level, we would be talking about “universal and existential quantifiers”; but even the basic language is tricky for the uninitiated.
Here is a typical question, from 2004:
Are Squares Rectangles? Are Rectangles Squares? Can a rectangle be a square? I just was wondering because a square can be a rectangle, but can a rectangle be a square?
Alisa’s comments suggest that she is confused about the word “can”, as I said above, but has no trouble with the concept of squares being rectangles. So I started by clarifying the wording, and the idea of classification:
Hi, Alisa. Actually, EVERY square is a rectangle, since the angles in a square are always right angles. That's more than saying that a square CAN be a rectangle; it IS. And since squares are rectangles, you know that some rectangles are squares--namely, the squares are! These ideas confuse many people. It may help to think about other categories, like dogs and animals. Can a dog be an animal? Of course -- every dog is an animal! Can an animal be a dog? Yes, if you look at a dog, that is an animal that is a dog. But not every animal is a dog. It's almost silly to ask and answer those questions.
Then I moved on to the other issue, which may or may not trouble Alisa: that classification uses words in a different way than everyday speech:
It sounds less silly with squares and rectangles, but the idea is the same. The trouble is that in everyday life, we usually use the word "rectangle" to refer to non-squares, because if the shape were a square, we would call it that. So we teach children "this is a rectangle, that is a square--the rectangle's sides are different lengths". But when you grow up, it's important to think of the square as a SPECIAL rectangle, because it is all that a rectangle is, and more. It's as if someone looked at your dog and said, "What a fine animal you've got!", and you answered, "That's not an animal--that's a pedigreed Labrador Retriever." The specific type of animal matters to you, because it's special; but it really is still an animal too. And a square is still a rectangle--but it's not JUST a rectangle, it's a (pedigreed) square.
Do you see the point here? We use words like “rectangle” or “dog” in two ways: Sometimes we want to specifically identify something (using the most precise term we can: “get me the square, not the mere rectangle”; “that is my dog, not just an animal”). On the other hand, sometimes we want only to put it in a general category (this square is a rectangle, because it has right angles; that dog is an animal, because it breathes). A rectangle is any four-sided figure with all right angles; a square is a particular kind of rectangle, namely one whose sides are also the same length. This is a more sophisticated perspective on the words.
Again, a dog (any dog) is an animal; an animal (some particular animal) can be a dog.
How can it be both?
Here is a 2006 question questioning the definition:
Classifying Quadrilaterals My teacher says that a square is also a rectangle. I don't understand how that can be, since the sides are the same length.
Adrienne’s concern goes beyond the meaning of “a square is a rectangle”, to the specific definition; her word “also” suggests that she thinks anything can have only one label. To her, a rectangle must be a figure whose sides have different lengths, so a square can’t be called a rectangle. I went a little deeper into the nature of definitions this time, but started with the same ideas (note how the old joke parallels my comment about the pedigreed dog):
In everyday usage, we would point to a square and say "that's a square, not a rectangle." That's because we generally name everything as specifically as possible. "That's not a lady, that's my wife!" In math and science, we have a slightly different perspective on words: we want each term to apply to anything for which it makes sense, and we want each definition to be as straightforward as possible. So, for example, we define a rectangle simply as "a figure with four straight sides, all of whose angles are right angles". If you compare a square with this definition, you see that it fits: it does have four sides, and it does have four right angles. So it is a rectangle. It's MORE than a rectangle, of course; in addition to the requirements of a rectangle, it also has four EQUAL sides. But that doesn't make it NOT a rectangle, only a more SPECIAL rectangle. We make definitions like this (called "inclusive definitions") for a reason. When we state a theorem about rectangles, we want to be able to apply it in every case where it is true. Since any fact about a shape that depends only on the fact that it has right angles will apply not only to "mere rectangles", but also to squares (special rectangles), it makes sense to use one word to cover them all, rather than having one word for "non-square rectangles", and another for squares. If nothing else, this makes it a lot easier to state theorems. The same is true in other fields of science. We use inclusive definitions in naming animals, for example: a terrier is a special kind of dog, and a dog is a special kind of mammal, and a mammal is a special kind of vertebrate. You wouldn't say "that's a terrier, not a mammal" just because "terrier" is a more specific term than "mammal"; we need to have a word that covers all mammals, so that we can talk about facts that are true of all of them. In the same way, a square is just a "species" of rectangle.
This is the key idea of classification: In order to create a hierarchy of categories, each has to include others. As a result, we tend to define things based on properties they have (e.g. right angles), not on properties they don’t have (e.g. equal sides). So everything that has specific properties is in the category, regardless of what else is true of it.
I continued, illustrating this idea with a diagram of the classification of quadrilaterals:
Just as we can make a whole classification tree for animals, we can classify shapes using these inclusive names. Here is a classification of the main types of quadrilaterals (four-sided figures): quadrilateral / \ / \ / \ kite trapezoid | / \ | / \ | / \ | parallelogram isosceles | / \ trapezoid | / \ / | / \ / rhombus rectangle \ / \ / \ / square Each figure is a special case of the figure(s) above it. Without inclusive naming, we would have to write theorems like "if you connect the midpoints of successive sides of a quadrilateral or kite or trapezoid or parallelogram or rhombus or rectangle or square, then the resulting figure will be a parallelogram or rhombus or rectangle or square." Using inclusive definitions, the theorem is just "if you connect the midpoints of successive sides of a quadrilateral, then the resulting figure will be a parallelogram." That saves a lot of trees!
For example, a rhombus has four equal sides; a rectangle has four perpendicular sides; a square has four equal, perpendicular sides, so it is both a rhombus and a rectangle — and also a parallelogram, and even a trapezoid, because any rectangle fits those definitions, too. [If you question this last statement, we’ll get there eventually!]
Here are two shorter answers, relating the square and the rhombus:
Definitions in conflict
It is important to recognize that both definitions are used, in different contexts; mathematicians generally use inclusive definitions, but elementary textbooks sometimes use the less technical exclusive definitions, presumably because they are aiming more at teaching everyday uses than at preparing for theoretical mathematics. Here is another question much like the others:
Inclusive and Exclusive Definitions Are squares rectangles? Are rectangles squares?
Doctor Floor answered this one, with an emphasis on potential disagreements:
Rectangles are not always squares, because squares need four sides of equal lengths. The question whether the other way around is true depends on the type of definition you are using. There are often disputes about definitions. In general there are two types of definition for geometric shapes: INCLUSIVE DEFINITIONS: In the case of rectangles and squares this means that a square is seen as a special case of a rectangle: * A rectangle is a quadrilateral with four right angles. * A square is a quadrilateral with four right angles and four equal sides. EXCLUSIVE DEFINITIONS: In the case of rectangles and squares this means that a square is NOT considered a rectangle: * A rectangle is a quadrilateral with four right angles but not four equal sides. * A square is a quadrilateral with four right angles and four equal sides. I prefer inclusive definitions, because they include the basic mathematical concept of 'generalization': one item (rectangle) is more general than the other item (square). But others say that exclusive definitions are very useful for special cases.
For a brief exploration of what actual dictionaries say about these definitions, see this discussion, where a teacher insisted on the exclusive definition:
Is an equilateral triangle isosceles?
Almost simultaneously, Michelle had asked a related question about triangles, which was archived along with that last one. She was troubled by the fact that different sources gave different answers:
I am in the 10th grade and I need to know whether an isosceles triangle has exactly or only two sides congruent. Is an equilateral triangle considered isosceles as well? I've looked at over 30 sites but I never get a full answer on whether there are exactly, only, or at least 2 sides congruent - or if I do, there is no explanation or reasoning to prove the statement.
If we define an isosceles triangle as one in which exactly (only) two sides have the same length, then it does not include equilateral triangles. This is the common way we think of it. But the inclusive definition is that at least two sides are congruent, so that it includes triangles with more than two sides of the same length.
Dr. Schwa answered Michelle’s concern, explaining why the inclusive definition is better in mathematics:
Hi Michelle, The question of whether an isosceles triangle has to have at least two sides congruent versus exactly two sides congruent isn't something you can prove: it's a question of definition. What does the term "isosceles triangle" mean? Definitions are something that we can choose arbitrarily, and books can have different definitions. However, some definitions are more useful than others, more convenient, easier to use... and in this case, one of those two definition choices is much better than the other. The inclusive definition is almost always better, as it is in this case. The inclusive definition, where isosceles triangles include equilateral triangles, is much more convenient. So isosceles triangles should be defined as triangles that have *at least* two sides congruent. Why is the inclusive definition more convenient? Well, consider a theorem like: If angle A = angle B in triangle ABC, then the triangle is isosceles. If you had the non-inclusive definition, you'd always have to be saying things like: If angle A = angle B but is not equal to angle C, then ... or If ... then the triangle is isosceles or equilateral. For the same reason, the definition of rectangle should include squares, the definition of parallelogram should include rectangles, and so on. The inclusive definition is almost always better.
Let’s look at one more discussion of this, from a teacher:
Is an Equilateral Triangle Also Isosceles? Can equilateral triangles also be classified as isosceles? Our 6th-grade math book defines an isosceles triangle as "a triangle with at least two congruent sides." The use of "at least" implies that an equilateral triangle could also be classified as isosceles, since it has "at least" two equal sides. (Perhaps in a similar manner to how a square is also a rectangle?) As a student, I had always learned that the three types of triangles, when classifying by side length (equilateral, isosceles, and scalene), were distinct and separate. An equilateral triangle has 3 equal sides, an isosceles has 2, and a scalene has different lengths for each side. Our school's text book has made me question this assumption, which is why I went to your site! I don't want to teach this incorrectly! Which is correct? Can an equilateral triangle also be thought of as isosceles?
This brings us right back to our old idea that we need inclusive definitions for theorems, but exclusive definitions when we need to identify something as precisely as possible, like our prized dog. Doctor Rick pointed out that this is a needed transition from the introductory use of the term to the more mature mathematical usage:
Hi, Karen. I definitely prefer the inclusive definition, using "at least". Yes, this is the same thing as saying that a square is a type of rectangle; an equilateral triangle is a type of isosceles triangle. I'm glad your textbook is going in this direction; it is the way mathematicians think. By the time the students take a geometry class, in which they learn to prove theorems, I would hope they would learn the inclusive definitions, because these are in keeping with the way theorems work. A theorem that holds for any rectangle will apply to a square, because a square fits the definition of a rectangle; there is no need to write a separate theorem for squares. Likewise, a theorem that holds for any isosceles triangle also applies to equilateral triangles, because they fit the definition of an isosceles triangle. So why not learn this way of thinking in sixth grade? That sounds good to me!
In another (unarchived) answer, Doctor Rick dealt with the question of what to do when a textbook uses the “wrong” definition:
However, the fact is that any decent math book will give its own definitions of terms, and the author's choice is law within the context of that book. If a book defines an isosceles triangle as a triangle with *exactly* two equal sides, that is the definition of an isosceles triangle whenever you find the word used in that book. Though we may prefer the other definition, we must accept the definition in use in the book, and understand everything that is said about isosceles triangles in light of that definition. That's one resolution to the confusion. I hope it helps.
This is an important thing to know about definitions: a mathematician is free to state how she will use a word, and such a “local” definition overrides any questions about what is “standard”. Of course, this will be done only when there is no generally accepted standard.
There is more we could say about definitions of shapes; in the future I will come back to this, looking at trapezoids and “diamonds”, among other issues.