![]() ![]() (And if I said I had a three-legged dog, you would know it has only three legs.) To say “but an ant also has four legs” would feel like a joke rather than a serious statement. Similarly, we are comfortable defining a quadruped as an animal with four legs, and don’t think that includes insects, with six. Mathematicians would tend to want to clarify, just to make sure they are entirely clear. The main context I can think of is in probability or combinatorics questions, as in “What is the probability that a poker hand contains three aces?” here there is some ambiguity, which may be resolved by the context. So, where do we need the word “exactly” to avoid ambiguity, and where does your student’s idea become mere pedantry? That’s hard to say. In this context, it is understood that the “3” mentioned is the total, not just a part. To say otherwise is to stretch the language too far. In my personal opinion, in the context of a definition, it is clear enough that “three” means “exactly three”. The surrounding words imply “exactly”, and adding “exactly” just emphasizes that. Reading these, it seems to me that “three-sided” means “having exactly three sides”, and “consisting of three segments” means “consisting of only three segments”. On the other hand, “ exactly” is sometimes used:Ī triangle is a polygon with exactly three sides. Triangle: A closed figure consisting of three line segments linked end-to-end. ![]() Every triangle has three sides and three angles, some of which may be the same.Ī triangle is a polygon with three edges and three vertices. In particular, it is clearly standard to say that a triangle is defined by having three sides:Ī triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. That’s an interesting point there are indeed times when a phrase like “three sides” can mean “at least three sides”, or “three sides, and perhaps others”, and “exactly” is needed for clarification but I don’t think that’s true here. I answered, tentatively, checking for what others say before giving my opinion: Yes, a quadrilateral has three sides … and another one, too! Does that count? Is it necessary to say “exactly three sides” when defining a triangle? Is it also accurate/correct to say that a quadrilateral has three sides? However, one student said a quadrilateral has also three sides. I defined a triangle as a polygon with three sides. This question came from a teacher, Mark, last October: ![]() Triangles: Must we say “exactly three sides”? We’ll be looking at the definitions of triangles (do we need to say “exactly three sides”?), rectangles (did Euclid use an exclusive definition?), and circles (can the circumference be called its length?). Several recent questions involved details about definitions of geometrical objects, so I thought I’d group them together, because each is relatively short. ![]()
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