2pi Radians as a complete revolution

[jpanknin]

I'm having trouble grasping why the angle measure in radians of a complete revolution of a circle is $2\pi$. If $\pi=\frac{C}{d}$ (also $\pi=\frac{C}{2r}$) then why does $\pi$ only cover half of the measure of a complete rotation and half the circumference of a unit circle? Seems intuitively like $\pi$ should measure the angle of an entire single rotation. I can work it out algebraically where if $C = 2r\pi$ then$\frac{C}{r}=2\pi$, but I'm trying to understand the intuition of why it's $2\pi$ as the measure of a full rotation instead of just $\pi$.

This is still a bit jumbled in my head, so happy to try to explain further if this is confusing.

 

[MarkFL]

It has to do with the way the radian, as an angular measure, is defined. It is equivalent to the length of a circular arc, where the radius of the circle is 1 unit, subtended by some angle. As the circumference of a circle is:

$\displaystyle C=2\pi r$

Then, 2π radians is 360°.

You will find the radian is a useful angular measure, making things much simpler.

 

[Dr.Peterson]

I'm having trouble grasping why the angle measure in radians of a complete revolution of a circle is $2\pi$. If $\pi=\frac{C}{d}$ (also $\pi=\frac{C}{2r}$) then why does $\pi$ only cover half of the measure of a complete rotation and half the circumference of a unit circle? Seems intuitively like $\pi$ should measure the angle of an entire single rotation.

Why? Just because we use the radius, not the diameter, as the unit of measure along the circumference. That's why it's called a radian.

One could define a unit in which $\pi$ units are a full rotation; but you might want to call that a "diametran" or something.

 

[Otis]

Seems intuitively like π should measure the angle of an [entire rotation] … happy to try to explain further if this is confusing

Hi jpanknin. Thanks. Please explain further why equating pi with one rotation seems intuitive for you. (I'm curious because the question arises regularly.)
$\;$

 

[jpanknin]

Thanks @MarkFL. I'm familiar with radians and with $2\pi$ AS DEFINED as the measure of a complete rotation of both the angle and the circumference (assuming a unit circle for the circumference). @Otis I think it's the $\frac{full\ Circumference}{full\ diameter} = half\ circle\ circumference\ and\ angle$. Why doesn't $\frac{full}{full} = full$? The book I'm using and the sites I've found online simply DEFINE a full rotation of a circle as $2\pi$, but don't give much intuition as how that number was arrived at. For example, $\pi$ is not DEFINED as 3.14...., rather 3.14... is the ratio of the circumference to diameter. So rather than using $\pi$ as a number that's DEFINED, we get the intuition as to how it was arrived at. CAPS meant for emphasis, not yelling.

 

[Dr.Peterson]

The book I'm using and the sites I've found online simply DEFINE a full rotation of a circle as 2π, but don't give much intuition as how that number was arrived at. For example, π is not DEFINED as 3.14...., rather 3.14... is the ratio of the circumference to diameter. So rather than using π as a number that's DEFINED, we get the intuition as to how it was arrived at.

Every source I know of defines the radian as @MarkFL did, not just as a full rotation being 2π radians:

It has to do with the way the radian, as an angular measure, is defined. It is equivalent to the length of a circular arc, where the radius of the circle is 1 unit, subtended by some angle. As the circumference of a circle is:

$\displaystyle C=2\pi r$

Then, 2π radians is 360°.

See, for example, here:

Radian - Wikipedia

en.wikipedia.org

One radian is defined as the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle. More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; ...​

They don't mention 2π until the next paragraph.

 

[blamocur]

My $0.02 worth on why I find ratio of arc length to the radius is somewhat more natural than ratio to diameter: when I look at a remote object I find it more natural to consider the ratio of its size to the distance. E.g., the Moon's size is approximiate 1/110 of its distance from Earth, so I think of its angular size as 1/110, not 1/220.

 

[Otis]

I think it's the $\frac{full\ Circumference}{full\ diameter} = half\ circle\ circumference$

Hi JP. Whoops, that equation looks like π = ½C, which is true, but only for unit circles! (Radian measure works with all circles.) It does seem like your intuition improperly associates rotations with pi in a one-to-one correspondence, but I'm still not sure why.

Regardless of the above, let's return to the radian unit-definition, referred to by Mark (i.e., in terms of arclength and radius) and see where that takes us.

Like π, the unit we call 'radian' is dimensionless and defined as a ratio of two lengths. That ratio is [distance along the circumference] to [radius], or:

Angle measure in radians = arclength/radius​

If the arclength subtended by an angle θ equals the radius, then the angle measures 1 radian.

θ = r/r = 1 rad​

​

​

How many radii fit in a circle's circumference? A little more than six. In other words, for a complete rotation (arclength=C), we have:

θ = C/r = (2π∙r)/r = 2π rad ≈ 6.28​

For half a rotation, the arclength is ½C.

θ = (½C)/r = (½)(2π∙r)/r = π rad ≈ 3.14​

So, a smidgen more than 3 radii fit within half a rotation.

Everything is proportional, so we're not restricted to unit circles. For half a rotation when r=5, the arclength is roughly 15.70796.

θ ≈ 15.70796/5 ≈ 3.14159​

θ = ½(2π∙5)/5 = 5π/5 = π​

Therefore, the reason why π rad corresponds to half a rotation (versus complete rotation) is simply because that's the way the radian unit HAS BEEN DEFINED (caps for emphasis, not shouting).
$\;$

 

[Dr.Peterson]

The book I'm using and the sites I've found online simply DEFINE a full rotation of a circle as $2\pi$, but don't give much intuition as how that number was arrived at.

It may be helpful if you show us an image of what your book actually says when it defines the radian. Possibly you are misreading something (though it is also entirely possible that the book is poorly written). If it is not in English, I'd want to see both the image and a translation.

The same is true for sites you've found. If you can show them to us, we may be able to help interpret what they are saying.

 

[jpanknin]

Thanks, everyone. I'm having trouble explaining what doesn't make sense when it doesn't make sense (if that makes sense).

For some context on why I'm digging on this, I already have two master's degrees (business related fields, not quantitative) and have gotten into an Ivy school for applied math and data science. I took one calc class and got my bu** kicked and realized my math experience consisted of memorizing formulas rather than genuinely and deeply understanding where these ideas come from, why they exist, and why they're important (I'm finding nothing in math exists without some deeper reason or as a relationship between other things and that's what I'm looking for). So I got Stewart's Algebra and Trigonometry (@Dr.Peterson, I doubt it's the book that's the problem) and have spent the last 6 months PAINSTAKINGLY going through it on my own and not moving on until I have the "a ha" moment. It's also been a lot of 9th grade math videos, which has been a bit hard on the ego.

Even small things like looking up the etymology of a math term or that a radian is defined as being equal to the radius helps my understanding (thanks @Dr.Peterson and @Otis for the figures) and allows me to "see" or "speak" the language of math.

So again, I'm having trouble understanding why it's not clicking at a deeper level that (assuming the unit circle) $\pi$ is half a circle's angle and circumference and $2\pi$ is a full circle and how this is also related to a radian. Perhaps it's that I somehow think that an even number of radians should be exactly a half circle and an even number of radians should be a full circle. Not sure, though. Perhaps it's not understanding how $\pi$ and radians are connected. Or maybe it's feeling that $\pi$ should be the measurement for a full circle angle and $\frac{1}{2}\pi$ should be the angle measurement of a half circle (i.e., one $\pi$ = full circle and one-half $\pi$ = one-half circle.

Hope this helps or at least doesn't make it more confusing.

 

[Dr.Peterson]

(@Dr.Peterson, I doubt it's the book that's the problem)

I didn't say that it is! I asked you to show us the definition that you are confused by, so we can talk about how to understand it. That would probably help a lot. At the least, there may be something about reading math texts that we could help with.

So again, I'm having trouble understanding why it's not clicking at a deeper level that (assuming the unit circle) $\pi$ is half a circle's angle and circumference and $2\pi$ is a full circle and how this is also related to a radian. Perhaps it's that I somehow think that an even number of radians should be exactly a half circle and an even number of radians should be a full circle. Not sure, though. Perhaps it's not understanding how $\pi$ and radians are connected. Or maybe it's feeling that $\pi$ should be the measurement for a full circle angle and $\frac{1}{2}\pi$ should be the angle measurement of a half circle (i.e., one $\pi$ = full circle and one-half $\pi$ = one-half circle.

This gives me the impression that you are letting preconceived expectations direct your thinking, and not following the definitions wherever they lead.

There are people who wish that instead of $\pi$, we had a special name for $2\pi$, in part for the reason you give, that it just feels more natural. Some of them call it tau ($\tau$); see here, for example. (It may not be irrelevant that this is from the April issue of a magazine.)

But that doesn't change the fact that (given what pi does mean) the circumference of a circle is $2\pi r$, and therefore the circumference of a unit circle is $2\pi$, and that is therefore the radian measure of a full turn (since the radian measure of an angle is the arc length on a unit circle).

 

[jpanknin]

I've attached an image from the book where radians and $2\pi$ as the circumference of a unit circle are first mentioned (this is from Stewart's Algebra and Trigonometry 4Ep. 438). This section mentions much of what was said above (and provided in the links). It talks about 1 radian being the arc length of the radius and defines the circumference and complete revolution angle measurement both as $2\pi$.

I guess it's this section that's not clicking for me in terms of how radians are measured and how $\pi$ and $2\pi$ are related. Just seems that $\pi$ and $2\pi$ are a bit arbitrary.

I do like the idea of tau as it does seem more intuitive having a single unit measuring the circumference and angle measure of a complete circle.

Thanks, everyone. I'll continue to look at this and see if something jumps out. Appreciate all the responses and help.

 

[Dr.Peterson]

I've attached an image from the book where radians and $2\pi$ as the circumference of a unit circle are first mentioned (this is from Stewart's Algebra and Trigonometry 4Ep. 438). This section mentions much of what was said above (and provided in the links). It talks about 1 radian being the arc length of the radius and defines the circumference and complete revolution angle measurement both as $2\pi$.

I guess it's this section that's not clicking for me in terms of how radians are measured and how $\pi$ and $2\pi$ are related. Just seems that $\pi$ and $2\pi$ are a bit arbitrary.

It's good to see that this book does indeed begin with the proper definition of radian measure, and then explains why that implies that 1 full turn is 2 pi radians.

But don't forget that most words and symbols we use in any kind of communication are essentially arbitrary; there may be a history behind it, but if you go back far enough, there's no logical reason we call a cow a cow, or whatever. In this case, it goes back to someone deciding to name the ratio of circumference to diameter rather than the ratio of circumference to radius.

 

[apple2357]

Is it also not case that if a radian was not defined in the way it was, results in calculus would not follow as neatly?
So for example the derivative of sinx is cosx but this is only true in radians. I think calculus drove the definition of a radian?

 

[jonah2.0]

Beer induced image contribution follows.

I've attached an image from the book where radians and $2\pi$ as the circumference of a unit circle are first mentioned (this is from Stewart's Algebra and Trigonometry 4Ep. 438). This section mentions much of what was said above (and provided in the links). It talks about 1 radian being the arc length of the radius and defines the circumference and complete revolution angle measurement both as $2\pi$.

I guess it's this section that's not clicking for me in terms of how radians are measured and how $\pi$ and $2\pi$ are related. Just seems that $\pi$ and $2\pi$ are a bit arbitrary.

I do like the idea of tau as it does seem more intuitive having a single unit measuring the circumference and angle measure of a complete circle.

Thanks, everyone. I'll continue to look at this and see if something jumps out. Appreciate all the responses and help.

Note: The so called Wrapping "Function" could be the ticket out of your disorientation.

 

[jonah2.0]

Beer induced ramblings contribution follow.

I'm having trouble grasping why the angle measure in radians of a complete revolution of a circle is $2\pi$. If $\pi=\frac{C}{d}$ (also $\pi=\frac{C}{2r}$) then why does $\pi$ only cover half of the measure of a complete rotation and half the circumference of a unit circle? Seems intuitively like $\pi$ should measure the angle of an entire single rotation. I can work it out algebraically where if $C = 2r\pi$ then$\frac{C}{r}=2\pi$, but I'm trying to understand the intuition of why it's $2\pi$ as the measure of a full rotation instead of just $\pi$.

This is still a bit jumbled in my head, so happy to try to explain further if this is confusing.

Perhaps another author's perspective could give you that aha moment.

 

[jonah2.0]

 

[jonah2.0]