Was going over math functions today in my MAT 115 class when I was asked the following question:
“Is the circumference of a circle a function of the radius? Or is the radius a function of the circumference?”
I didn’t have a good answer.
There’s no reason to assume there needs to be exclusivity. They can be functions of each other.
f® = 2pi*r
f© = c/2pi
What** friedo **said. There is a relationship between the radius and the circumference where you can determine one as a function of the other.
I suppose that in formal terms you can define one in terms of the other, and which end you start from is just a matter of choice or convention. However, surely the standard and simple definition of a circle is in terms of its radius and its center. A circle is the set of all points in a plane that are the distance r (the radius) from a given fixed point (the center). I suppose it must be possible to define a circle by reference to the circumference without mentioning the radius, but surely any such definition is bound to be a good bit more complicated. (Can it even be done without mentioning pi? And can pi be defined without mentioning radii or diameters?)
On that basis, I would say the radius is more fundamental.
You can substitute the radius for the sum of the square of the sides of a right triangle that has that radius as its hypotenuse, which puts the circle in terms of Cartesian x,y coordinates rather than a polar coordinate, namely, x^2 + y^2 = 1.
You can dispose of pi by using a different constant, which is twice the size of pi, called ‘tau’ and stop dealing with diameter and just the radius, which is the true function of defining the circle. See here.
Yes.
Each is the inverse function of the other.
The length of the radius can be any positive real number x. There is a one-to-one function, namely y=2Pi*x, that maps any positive real to the positive reals. That means that there is an inverse function, namely x=y/2Pi, that maps y back to the original x.
The length of the circumference can be any positive real number x. There is a one-to-one function, namely y=x/2Pi, that maps any positive real to the positive reals. That means that there is an inverse function, namely x=2Pi*y, that maps y back to the original x.
It’s completely up to you which function you pick as the “original,” and which as the inverse.
If the instructor expects a “unique” correct answer, I’d go with radius. Why? Because a circle is usually defined as a set of all points equidistant (with distance r) from a single point. In this way, the circumference is the total length covered by these points, whereas the radius is the parameter that defines where these points are relative to the “original point” (center of the circle).
But as mentioned, either can be defined mathematically in terms of the other with no problem.
Sorry that should be “I’d go with circumference is a function parameterized by the radius”, I realize now that wasn’t as clear as I meant it to be.
A circle could be defined as the shortest possible closed curve enclosing any specific area (granted, that isn’t a recipe for drawing one).
Somehow I missed njtt’s post the first time through the thread :smack:
Anyway, while I’m certain you can concoct a definition involving the circumference like that, the standard definition given in most geometry classes is still probably “a locus of all points equidistant to a single point.”* So ‘traditionally’ I’d say radius is the more ‘basic’ variable, with the scare quotes to emphasize the fact that it’s still arbitrary.
- As I recall, that was word for word the definition in my geometry class. Not often you get to use the word “locus”, it makes me feel fancy.
It could be defined this way, but it’s not.
Well, it is, in a hair-splitting way. Just like every one of the bazillion statements in the Invertible Matrix Theorem is a complete and working definition of an Invertible Matrix. It’s just not the most commonly used definition.
Or better, if we’re going in terms of the circumference, then it’s the curve of a given length (the circumference) that encloses the maximum possible area. I think this generalizes to a definition for a 3-sphere (surface of a given area that encloses the maximum volume), and higher dimension n-spheres, as well. Though I don’t think it works for for non-Euclidean curved spaces. But then again, the standard definition of Pi doesn’t work on non-Euclidean curved surfaces, so maybe we can ignore them.
And Pi pops up in so many places, it’s easy to make a definition of Pi that doesn’t involve circles in any way, though I can’t remember any off the top of my head (isn’t there an easy formula involving the sum of an infinite series?)
It * can be*, so it is.
Yea, my answer was that circumference can be a function of radius and vice versa. But then we got into a chicken or the egg discussion, and I opined that it might be more correct to say circumference was a function of the radius for the same reason you gave.
But not both: that would be circular.
Well, I think you are making my point for me. Defining the circle without using the radius is more complicated.
Whether you use tau or pi does not really matter, but whichever you choose, if you use it in in your definition you are going to need a way of defining pi (or tau) that does not make reference to circles. That may be doable, but it is bound to be complicated.
OK, I guess that is a simple definition in terms of circumference, but, as you intimate, it is not actually very useful for doing geometry. I think there are good reasons why the definition in terms of radius and center is the standard one.
Yes.
Yes.
Understandable. It’s analogous to the question “Is an apple a fruit? Or is an orange a fruit?”
You can have a separate discussion about which is more “fundamental”.
Incidentally, in engineering it’s usually much easier to measure the diameter than the radius, so expressing circumference and area as a function of radius is not very useful.
Sure it is. You even cited an example of it being done.