# Why do we have circles for ellipses, squares for rectangles but nothing for triangles?

migrate to math ed se if need be please. i'm lazy =)

A circle is an ellipse with equal foci. A square is a rectangle with equal sides. Why is there no special name for equilateral triangle?

Context is

1. people develop misconception in kindergarten that squares are not rectangles

But why in the first place do we have special names for certain types of ellipses and rectangles but don't have special names for equilateral, isosceles or right triangles?

CMV: I think everyone's lives would be much easier if the words 'circle' and 'square' did not exist. To describe a rectangle with equal sides, we follow what we do with a triangle with equal sides:

1. For kindergartners or anyone who doesn't know the word 'equilateral', call them rectangles just as is done for triangles i.e. no distinction made between an equilateral triangle and a non-equilateral triangle

2. For people who know the word 'equilateral', replace 'square' with 'equilateral rectangle'

But I'm guessing there may be something to do with quadrilaterals since there are a lot of possible quadrilaterals relative to possible trilaterals

• Isn't a scalene triangle the "rectangle" of equilateral triangles? Apr 24, 2017 at 13:58
• @Justin Squares are rectangles while equilateral triangles are not scalene triangles? I think scalene triangle is analogous to oblong / re 'Isn't a scalene triangle the "rectangle" of equilateral triangles?'
– BCLC
Apr 24, 2017 at 14:02
• Circles are fundamental building blocks of classical geometry: they are the "compass" part of compass-and straightedge constructions. Ellipses are not. Apr 27, 2017 at 3:52

This is a question about linguistics and psychology and teaching, not really about mathematics.

We have special words for things we refer to often. Circles come up way more often than ellipses so it's convenient (and clear historically) that they have their own word. "Square" is much nicer than "equilateral rectangle" and requires a lot less cognitive processing.

I spend a fair amount of time in K-5 classrooms, so I've some experience with the questions you raise.

Yes, kids in elementary school are confused by the fact that a square is a rectangle. So are some elementary school teachers. That's a problem with trying to impose correct formal mathematics on informal everyday speech. It happens a lot - this is an instance (in a way) of whether "or" means "and or" or "or but not and". In mathematics it's always the former. In daily life, sometimes one sometimes the other.

One problem I have with elementary school "geometry" is its focus on categorizing and naming things and its paucity of theorems - or at least observations of properties. I wish kids were taught to notice that the diagonals of a parallelogram bisect each other, or that the medians of a triangle meet at a point, way before they encounter proofs.

And yes, this should be migrated to math education SE.

• With an eye towards the 'categorization' aspect, it's also true that the number of categories for a triangle are a lot narrower than for a quadrilateral. In the former case, a triangle can either be categorized according to its lengths (scalene, isosceles, or equilateral) or according to its angles (acute, obtuse, or right). These are enough to categorize a given triangle. But the number of variations for a quadrilateral goes up considerably: rhombus, parallelogram, rectangle, square, and so forth. So it seems almost inevitable that a profusion of names exists in the latter case. Apr 24, 2017 at 14:22
• @Semiclassical Don't forget the trapezoids. And it's even worse than you suggest. Quadrilaterals can be convex (or not) and self-intersecting (or not). Also degenerate (colinear), as can triangles. I discuss these things with teachers, who then decide whether to talk about them with (some of the) kids. Apr 24, 2017 at 14:28
• Thanks Ethan Bolker. Okay more cognitive processing for equilateral rectangle but why don't we just say rectangle? Why emphasise that the sides are (approximately?) equal? I mean, what real life situations are there s.t. we 'square' or 'circle' is much better than, resp, 'rectangle' or 'ellipse' ? P.S. I think perfect is a pretty good sub for equilateral?
– BCLC
Apr 24, 2017 at 16:36
• @BCLC Kindergarten kids are likely to know the one syllable words "square" and "circle" before "rectangle" and certainly before "ellipse". That's what you build on. The formal distinctions later are as much about learning to make language precise when necessary and about sorting using attributes as about the geometry as such. I'm sure there's lots of literature out there about elementary education that addresses these questions. I'm just an observer. Apr 24, 2017 at 17:41
• @BCLC Circle is still right for several reasons. If you used "oval" for what we now call "circle" would the circle be a "regular oval" or would it be the name for circle, keeping ellipse for the more general figure? I don't think you should create a sudden disconnect between what kids know (vocabulary and experience) coming into school with the language of mathematics. Do that gradually, as necessary.. You're not going to change the language by decree to make it simpler (or more mathematically precise). Think Esperanto. Apr 25, 2017 at 12:55