how do transcript 172.70.127.37 19:23, 10 March 2023 (UTC)

https://en.wikipedia.org/wiki/Square_degree may be of some help with this one. 162.158.166.124 19:44, 10 March 2023 (UTC)

The comic isn't actually correct. A radian is not equal to the length of a circle's radius; it is equal to the length of the radius, multiplied by 2π, divided by the perimeter, which is why it has no units, while the length does. In other words, radian/2pi=length of radius/length of perimeter. 172.70.46.84 19:51, 10 March 2023 (UTC)

As suggested by the above Wikipedia link, square degrees are in fact often used in astronomical contexts. Also, it's quite standard to say that radian=1; see for example SI derived unit. An angle is the ratio between the arc length and the radius, and we just optionally append "radian" for clarity. So 1 = 57.3 degrees is correct; Randall simply used the wrong argument to obtain it. Aseyhe (talk) 20:57, 10 March 2023 (UTC)

I always understood radian to be the name of the unit, so by definition 1 radian=1. Barmar (talk) 21:17, 10 March 2023 (UTC)
It is a shame that astronomers don't use the proper unit for such things: the steradian. It is literally there for describing the 3D equivalent of angle. Oh well... --172.69.79.137 04:16, 11 March 2023 (UTC)
It is a shame that astronomers don't use the proper for length, preferring ad-hoc units based on the solar system. But if you use a different ad-hoc unit based on the properties of the solar system they throw a hissy fit.172.70.38.150 06:51, 12 March 2023 (UTC)
Indeed, what is the "proper [distance unit?] for length"? Light-year, based on Earth's orbital period. AU, based upon Earth's orbital radius. (Kilo)metre, based (approximately, and quartered) upon Earth's circumpolar circumference. Parsec, based upon Earth's orbital radius and a notionally arbitrary subdivision of angle. (Which can be avoided by mathematically more pure "paradians"???) Planck-lengths, might be not solar-/geo-centric but creates horribly huge numbers even at the human scale. ;) 172.70.86.128 16:07, 12 March 2023 (UTC)
Planck length could work. Large Number problem can be resolved by 10insert number here Planck Lengths, since astronomers already do it, and round _insert number here_!! 1844161 (talk) 15:43, 14 April 2023 (UTC)

Someone fix the vandalism, how do you upload images? --Purah126 (talk) 03:06, 11 March 2023 (UTC)

I'm doing it but that user needs to be blocked.
To revert images, scroll down and click the revert link next to the last good version.
And do not feed the trolls. ~ Megan she/her talk/contribs 03:10, 11 March 2023 (UTC)

On reading this I vividly remembered a maths teacher once asking our class "What's 10% of a straight line?", and the looks of disgust and bewilderment when he said the answer was 18 degrees. 172.70.86.147 08:31, 11 March 2023 (UTC)

I just hope that was Celsius degrees (or Kelvin), rather than Fahrenheit(/Rankine). ;) 172.71.242.190 10:51, 11 March 2023 (UTC)
If you use Kelvin with degrees you have already lost...172.68.51.178 13:29, 11 March 2023 (UTC)

So the volume of the sky is 4/3 π r³ = 7,092,429 cubic degrees

I remember in the quantum mechanics class we figured that if \hbar is defined to be h/2π, then we might as well introduce the notation \pibar as an alternative for 1/2. Captain Nemo (talk) 11:08, 12 March 2023 (UTC)

The logic is fine once you recall the formula s = r x theta. The arc length subtended by an angle is equal to the radius times the angle. On the unit circle, the radius is 1 (no unit). Therefore, the subtended arc length of 1 radian is s = 1 x 1 radian = 1 radian. 172.71.22.117 21:45, 12 March 2023 (UTC)

"...the radius is 1 (no unit)." There's definitely a unit. It's whatever the unit the unit circle is reflecting (even if that's mathematical Unity). And in the case of dimensional analysis, it's a particular dimension that you'd need to account for, and the difference between this radians thing and the degrees thing is only the inclusion of dimensionless pi-based constant of conversion. Doesn't change the understanding of the issue, but I believe that some explanations/comments aren't then conveying it onwards accurately. 172.69.79.184 22:15, 12 March 2023 (UTC)
I mean, I'm sorry, but respectfully, you are wrong. The unit circle is *by definition* a circle of radius 1. There is no unit attached to that. 172.71.82.41 01:55, 13 March 2023 (UTC)
Correction: The unit is that of the radius, by definition. It is one of that unit, whatever that unit may be. You attach whatever unit you want to it, when you want to, but it isn't actually a unitless value when you start comparing it with othe values whose relationship and own unit are known. 172.71.178.207 03:59, 13 March 2023 (UTC)
I'd say the radius of a plain unit circle is unitless, but not dimensionless. It has a length dimension, but we don't necessarily attach a unit to that dimension. Pmc (talk) 18:19, 25 April 2023 (UTC)

There is actually some dispute about whether angles should be measured using units. I can't find it now, but there was an article by someone arguing that the current SI definition of the radian as 1 rad = 1 m / 1 m was flawed. He felt that units of angle should have a dimension, A, and rewrote several formulae slightly to accommodate this. But more often today, the radian is considered dimensionless with a value of exactly 1, making it not actually a "unit" so much as a hint telling how the angle was measured. In this definition, an angle has a measure of x (radians) iff the circular arc it intercepts as a central angle has an arclength of x times the circle's radius. Under this definition, the following become mathematically correct:

° = π/180
Radius of unit circle = 1 = (180/π)(π/180) = (180/π)° = 57.29577...°
(1°)² = π²/32400