If you take a circle to be the limit of a polygon as the number of sides goes to infinity, then you have infinite interior angles, with each angle approaching 180deg, as the edges become infinitely short and approach being parallel. The sum of the angles is infinite in this case.
If you reduce this to three sides instead of infinite, then you get a triangle with a sum of interior angles of 180deg which we know and love.
On the other hand, any closed shape (Euclidean, blah blah), from the inside, is 360deg basically by definition.
Except that the angle of a circle’s circumference is measured as an arc with the vertex at the center, and to include an infinite number of angles you would need to reduce the degrees accordingly to avoid overlapping
That’s exactly my point, there are two different colloquial ways of talking about angles. I am not claiming there is a mathematical inconsistency.
Colloquially, a “triangle has 180 degrees” and a “circle has 360 degrees.” Maybe that’s different in different education systems, but certainly in the US that’s how things are taught at the introductory level.
The sum of internal angles for a regular polygon with n sides is (n-2)×pi. In the limit of n going to infinity, a regular polygon is a circle. From above it’s clear that the sum of the internal angles also goes to infinity (wheres for n=3 it’s pi radians, as expected for a triangle).
There is no mystery here, I am just complaining about sloppy colloquial language that, in my opinion, doesn’t foster good geometric intuition, especially as one is learning geometry.
I see. That almost makes sense, but pi radians = 180°
Also, the value of one internal angle of a regular polygon is (n-2)×(π÷n), in which case π÷n is infinitesimally small. In other words, substituting infinity for n would be incalculable, and even if it were, adding them together wouldn’t equal infinity because the larger n is, the smaller each individual internal angle.
It’s not about colloquialism or language, there are immutable principles of geometry, and adding the internal angles of a triangle gives you 180°, whether you express it as such or as π radians or 3200 mils or something completely different doesn’t matter. That’s just changing the unit of measurement but the underlying principle is the same.
Circles can be confusing and counterintuitive, but that’s why they need an irrational number in order to be expressed. If you’re measuring the internal angle you’ll probably express it as an arc, because infinite and infinitesimal numbers are impossible to express rationally.
Take for instance, calculating angular momentum with a circle. You have to calculate it based on the tangent because the circle itself doesn’t give you any constancy otherwise.
I don’t think we’re talking about the same thing.
If you take a circle to be the limit of a polygon as the number of sides goes to infinity, then you have infinite interior angles, with each angle approaching 180deg, as the edges become infinitely short and approach being parallel. The sum of the angles is infinite in this case.
If you reduce this to three sides instead of infinite, then you get a triangle with a sum of interior angles of 180deg which we know and love.
On the other hand, any closed shape (Euclidean, blah blah), from the inside, is 360deg basically by definition.
It’s just a different meaning of angle.
See, for example, the internal angle sum, which is unbounded: https://en.wikipedia.org/wiki/Regular_polygon
Except that the angle of a circle’s circumference is measured as an arc with the vertex at the center, and to include an infinite number of angles you would need to reduce the degrees accordingly to avoid overlapping
That’s exactly my point, there are two different colloquial ways of talking about angles. I am not claiming there is a mathematical inconsistency.
Colloquially, a “triangle has 180 degrees” and a “circle has 360 degrees.” Maybe that’s different in different education systems, but certainly in the US that’s how things are taught at the introductory level.
The sum of internal angles for a regular polygon with
nsides is(n-2)×pi. In the limit of n going to infinity, a regular polygon is a circle. From above it’s clear that the sum of the internal angles also goes to infinity (wheres for n=3 it’s pi radians, as expected for a triangle).There is no mystery here, I am just complaining about sloppy colloquial language that, in my opinion, doesn’t foster good geometric intuition, especially as one is learning geometry.
I see. That almost makes sense, but pi radians = 180°
Also, the value of one internal angle of a regular polygon is (n-2)×(π÷n), in which case π÷n is infinitesimally small. In other words, substituting infinity for n would be incalculable, and even if it were, adding them together wouldn’t equal infinity because the larger n is, the smaller each individual internal angle.
It’s not about colloquialism or language, there are immutable principles of geometry, and adding the internal angles of a triangle gives you 180°, whether you express it as such or as π radians or 3200 mils or something completely different doesn’t matter. That’s just changing the unit of measurement but the underlying principle is the same.
Circles can be confusing and counterintuitive, but that’s why they need an irrational number in order to be expressed. If you’re measuring the internal angle you’ll probably express it as an arc, because infinite and infinitesimal numbers are impossible to express rationally.
Take for instance, calculating angular momentum with a circle. You have to calculate it based on the tangent because the circle itself doesn’t give you any constancy otherwise.
Right, a triangle “has 180deg,” like I said.
That’s not how limits work. Substitution is not the same as taking the limit.
That’s not true at all. https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_⋯
Having one word (or phrase) with two meanings is a property of language.