Talk:Angle

Turn is period?
Reddwarf2956 has been making some improvements to this article (which I support), but we seem to disagree on the definition of a turn. I don't understand how it can be a period. Am I missing something?  D b f i r s   07:29, 15 September 2013 (UTC)
 * A circular movement is periodic, and the turn is its period. Nevertheless this concept seems too technical for this elementary article, and it is better to avoid to mention "period" here. However, I agree that this paragraph is not satisfactory, as giving a circular :-) definition. Therefore, I suggest "the turn is the angle for which the above circular arc is the complete circle". D.Lazard (talk) 09:53, 15 September 2013 (UTC)
 * I agree that there might be some loose association of a turn with a period in some usage, but I still don't see how a turn can be a unit of time. I agree with your suggested improvement.    D b f i r s   06:25, 18 September 2013 (UTC)

Transversal and polygons both have interior angles and exterior angles
Comments at the article edit: "I don't see what the association is for polygons. No mention at the linked article." See Angle. Let's add to the current statements and move them to in a different section of angle. But, we still need some references to both interior angles and exterior angles of Transversal_(geometry) with the Transversal section in angle and with polygons. Maybe combine the transversal and polygons sections together and define the terms?

John W. Nicholson (talk) 02:51, 23 September 2013 (UTC)

Small article merger
I am trying to combine as many of the small articles on different angles into angle (per talk), but I realize that some may not fit very well. Those like transversal, polygon, and both interior angles and exterior angles don't seem to fit currently, and wiki currently has separate articles. Others like Adjacent, Complementary, Supplementary, and Vertical angles seem to easily fit. Please take look at these and decide if any more parts should be moved to this article, angle, as to keep the information or start a AfD for each. Feel free to start a AfD if you feel that it's good. — Preceding unsigned comment added by Reddwarf2956 (talk • contribs) 21:25, 24 September 2013 (UTC)
 * RockMagnetist used a template at 05:06, 20 November 2013 to document nicely, within the initial section (consisting primarily of "header templates") on this talk page, their own merging into the accompanying article page of the 4 (small) angles articles that RD had alluded to as "seem[ing] to easily fit". (I would infer, from RD's (mistaken) ref to AfD that RD did 4 merges, so perhaps RM re-merged, believing there had earlier been a problem with each.)   Of course RD deserves at least our gratitude for the valuable work of calling attention to the opportunity, while (by the way) knowing, for use in this talk section, neither
 * the pipe-trick markup that lets an editor avoid, e.g., a link to the Dab'n of the term "transversal", by linking instead to the actual article "Transversal (geometry)", yet displaying "Transversal", and
 * what you might call "the suffix trick" -- eliminating the last blank in the markup for "Internal and external angle s" to get, voilà, "Internal and external angles", thus avoiding the misleading impression that "interior angles and exterior angles" invites: that WP regards external angles as "the mere bastard siblings" of internal angles.
 * --Jerzy•t 07:16, 21 July 2014 (UTC)
 * Note the date stamps: I did the merges a few months after the comments by Reddwarf2956. RockMagnetist (talk) 15:41, 21 July 2014 (UTC)
 * That made no sense to me on my first several readings, and i responded at User_talk:RockMagnetist, bcz i could not see why RM thot (either or both) that i would care when the merges were made, and/or that i didn't already know. RD's confusing attempts to use wiki-markup to talk about articles, creating the three irrelevant and misleading links "Adjacent", "Complementary", and "Supplementary", may have been a stumbling block for either RM or myself; in any case, i've been at this too long, and not fruitfully enuf, to continue without a generous break. --Jerzy•t 10:14, 25 February 2017 (UTC)

Definition?
The definition of angle on the main page seems rather vague. Perhaps a better definition would be: the fraction of the arc of a circle with a center at the origin of the angle.

That way degrees can be clearly defined as:

(s/c)&times;360

and radians as:

(s/c)&times;2π = s/r

where s = arc length, c = circumphrence, and r = radius. '' Evidently unsigned, but from about late 2001; see Jerzy's contrib below which begins "The initial contrib ..." . ''
 * I agree that the definition in the first paragraph of the main page is vague, and in fact inaccurate because an angle is not necessarily bounded and so cannot be defined as a figure; defining them in terms of measure as the second paragraph does is a better approach. However, I don't agree with this comment in defining an angle in terms of a circle because while angles exist in a context that is essential to the specification of their measure definition, angles are not necessarily part of a circle. Skylarkmichelle (talk) 19:47, 19 December 2013 (UTC)
 * "Angle" has two different meanings. It is first a geometrical figure (or shape). As such, the first paragraph is not vague (the figure is precisely defined). It is not only accurate, but also correct, which is important. Note that this definition dates to Euclides, more than 2000 ago.
 * "Angle" is also a measure. As a measure, there are many equivalent ways to define it, which involve either circles or trigonometry. The definition given here seems to be the simplest. It says implicitly that, to measure an angle (figure) in which there is no circle, one has to draw a circle centered at the vertex of the angle and to measure the ratio of the length of the arc delimited by the figure and the radius of the circle. D.Lazard (talk) 20:37, 19 December 2013 (UTC)
 * I want to endorse the thrust of DL's comment, while saying that IMO it's not precise enuf for the article -- a choice that i not only respect, but probably agree with. --Jerzy•t 16:30, 20 July 2014 (UTC)
 * The initial contrib to this section joined the page between 17:54, 30 August 2001 and 15:43, 25 February 2002, i.e. approximately late in the first year of WP; this section's heading, at least, was retrofitted later. (The "user" shown in the edit history, user:Conversion script, was a special piece of code that i've never bothered to acquire any substantive knowledge of, but clearly was used in the very early days -- on many, perhaps all, of  the pages that then existed -- probably in order to add or improve an important, but presumably relatively obscure, design feature in the early wiki engine. It may be that no record of even the IP address of the contributor was ever made. In short, and for certain, i cannot determine anything beyond its being contributed to WP between day 1 -- very early in 2001 -- and the 2002 date.) --Jerzy•t 16:30, 20 July 2014 (UTC)

talk page destruction!
Who eviscerated this talk page? Yesterday it had 36 sections and today it has 4?!! Skylarkmichelle (talk) 22:43, 20 December 2013 (UTC)
 * The discussions have been Archived but are still available for reading. At the header of this discussion page, click Show in the Archive bar. Then click on the numeral 1 that appears at the left. You can refer to issues that have been raised, but bring your discussion to this current Talk. It is common practice to Archive old discussions from lengthy talk pages.Rgdboer (talk) 03:14, 21 December 2013 (UTC)

Proposed merge
I had been contemplating merging the article Internal and external angle into the section Polygon related angles here since this section was better written and there did not appear to be much chance of expanding the current article. I would have done this myself, but a recent spate of activity suggests that I had better ask for some opinions on this merge. Bill Cherowitzo (talk) 18:43, 4 October 2014 (UTC)


 * Support, with sources added. Otherwise delete. - DVdm (talk) 19:21, 4 October 2014 (UTC)
 * weak Oppose - looking at this article leaves with a somewhat dissatisfied feeling as it mixes imho to many things which might be clearer to various readers if they are structured differently and distributed over several articles. That is in particular separate different levels of mathematical approaches from notational, engineering or unit related material. In addition the current mathematical part somewhat switches unstructured between rather abstract notions (without any motivation) and more elementary ones. We would need an angle treatment on highschool level (comprising angle between lines, 2d and 3d vectors, lines and planes, between planes), then the generalisation to curves and surves, more than 3 dimensions and non euclidean spaces. This could be handled in one large well structured article but imho for accessibilty/readabilty reasons 2-3 articles might be better. In any case those should be separated from the other (non-math) content in the current article. Since I favour the splitting and restructuring (as well as extending the splitted parts) of this article, I'm somewhat opposed to merging additional stuff into this article.--Kmhkmh (talk) 18:34, 7 April 2015 (UTC)
 * weak Oppose and conditional strong oppose: This topic is important in the history of mathematics. In addition, I can see this article being expanded in the future to describing how various cultures throughout history have measured angles (this is Ethnomathematics), which would help in countering Systemic bias. I think that articles on English Wikipedia that deal with ancient topics in mathematics are consistently written with a greater focus on mathematics in the western world than elsewhere so if this article to expanded to include more mention of how throughout history other cultures have measured angles then I would say strong oppose. Mgkrupa (talk) 16:23, 3 September 2020 (UTC)

Supplementary & Complementary
These terms are usually used to refer to pairs of angles rather than to sets of three or more. A South African editor believes that the terms are applicable to larger sets of angles (and I asked the same question here several years ago). The OED specifies "pair of angles", but is there a reference for more general usage? (If there is, then I apologise for my reversion of the recent edit.)   D b f i r s   07:05, 14 July 2016 (UTC)

It's all in the reflexes
What about reflex angles? — Preceding unsigned comment added by 75.139.254.117 (talk) 17:58, 24 November 2016 (UTC)
 * Reflex angles are defined in the article, and explained with a figure. The phrase "reflex angle" appears 6 times. What do you want more? D.Lazard (talk) 18:05, 24 November 2016 (UTC)

Merger of Angular unit into Angle
they overlap widely. fgnievinski (talk) 05:09, 7 May 2021 (UTC)
 * Oppose: they are in appropriate Summary style . Klbrain (talk) 20:00, 22 May 2021 (UTC)
 * Support: Angular unit has a lot of material about angles and their measurement that does not belong there, and little detail about units that is not already at Angle. It does not merit a separate page in its current form.  —Quondum 20:15, 22 May 2021 (UTC)
 * Support per nom. In reply to, there is not enough unique and encyclopedic material in Angular unit to merit invoking WP:SUMMARY. &mdash; Cheers, Steelpillow (Talk) 17:40, 9 July 2021 (UTC)
 * All fair points, which I'm happy to concede. No objections to a merge from me. Klbrain (talk) 22:57, 9 July 2021 (UTC)
 * ✅ Klbrain (talk) 15:40, 1 August 2021 (UTC)

"measure"?
The statement
 * Angle is also used to designate the measure of an angle or of a rotation.

is meaningless as it stands. A measure is only defined over a set, which is not given: an angle is simply defined here as "the figure formed by two rays [...] sharing a common endpoint". To use a concept from measure theory, even if it were properly done, is totally inappropriate for an article such as this, especially in the lead where every word should be easily understood (and not easily misinterpreted) by a high-schooler. The subject matter of this article is not too complicated to explain in everyday words. I suspect that the average reader will misinterpret the word measure to mean measurement due to the similarity of the words. Note also that angle is naturally interpretable as a signed quantity, especially in relation to a continuous rotation, which makes measure an especially poor fit. Nowhere in the article is a measure used to define the angle as a quantity.

I suggest that any reference to measure theory should be omitted. It is neither needed nor used in the article. —Quondum 13:03, 24 May 2021 (UTC)
 * It is only a wrong link, probably added by someone that searched in sections of Measure (disambiguation) without remarking that this is the primary meaning that applies here. I'll fix it. D.Lazard (talk) 17:07, 24 May 2021 (UTC)
 * That was a definite improvement. Wouldn't a more common and natural term be "size of an angle", as opposed to "measure of an angle"?  —Quondum 17:19, 24 May 2021 (UTC)
 * This is a question for a native English speaker. My opinion is that "size" is not convenient here. For example, an angle may have a negative measure, and a size is generally supposed to be positive. Also, "size" is generally qualitative, and "measure" is quantitative. For example, the area and the diameter are two different ways to measure the size of a plane figure. D.Lazard (talk) 17:55, 24 May 2021 (UTC)
 * Fair points. I can't think of a word that works better while addressing what you mention, and it works adequately as is, so I'll leave it be at this stage.  —Quondum 18:16, 24 May 2021 (UTC)

"Ángulo" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Ángulo and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 February 14 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. User:1234qwer1234qwer4 (talk) 06:11, 14 February 2022 (UTC)

"Ângulo" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Ângulo and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 February 14 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. User:1234qwer1234qwer4 (talk) 06:11, 14 February 2022 (UTC)

Proposed merge of Angle of rotation into Angle
Joyous! | Talk 22:55, 19 February 2023 (UTC)

"Angle of rotation" is already covered in Angle. fgnievinski (talk) 03:43, 29 March 2022 (UTC)
 * Support Angle of rotation is just an unsourced stub, there is no benefit to having it separate. --Mathnerd314159 (talk) 18:07, 20 May 2022 (UTC)
 * Support – though it might make sense to delete it and consider leaving a redirect to Rotation instead of to Angle, since an angle might or might not parameterize a rotation. —Quondum 13:10, 19 October 2022 (UTC)
 * Support the alternative target, Rotation. It would be well placed there, perhaps before Rotation which explicitly uses the angle of rotation without describing it in one dimension. Klbrain (talk) 10:08, 2 January 2023 (UTC)
 * This is a bad idea. If someone wiki-links to angle of rotation they are looking for a definition of an angle measure, not a definition of a rotation. –jacobolus (t) 19:45, 2 January 2023 (UTC)
 * Good point; I had overlooked that common use. So, switching to support to the original target, Angle. Klbrain (talk) 08:43, 7 January 2023 (UTC)


 * Support it's a bit silly for this to be a separate article. It's a simple topic, one of the most basic expressions of an angle.

Looks like everyone is in solid agreement. Courtesy pinging. I guess I could do the merge, but I guarantee you'd want someone else, just to avoid the furrowed brows as everyone tries to puzzle out "Now why did she put it there???" Joyous! | Talk 22:55, 19 February 2023 (UTC)


 * ✅ —Quondum 01:41, 23 February 2023 (UTC)
 * Thanks for the merger done sometime ago. I've recently sourced a definition for angular displacement from an authoritative source (ISO 80000-3) which states explicitly it may be a signed quantity. Therefore, I've retargeted the redirect Angle of rotation. fgnievinski (talk) 08:52, 22 July 2023 (UTC)

Math
Like what is explanation of angles 185.76.177.40 (talk) 08:41, 6 May 2022 (UTC)

Dimensional subtlety
In "the measure of the angle in radians", I interpret this to mean an angle, expressed in the unit radians. For comparison, if I ask for someone's height in metres, I expect to get an answer like "1.83 m". That is, I am not asking for their height divided by one metre, which would be "1.83", but that the unit in which the answer is to be expressed be the metre. I could say that "Their height (in feet) is 6 ft", but to say "Their height in feet is 6" sounds wrong. Thus, the "the measure of the angle in radians" is an angle quantity and includes the unit, and is not a bare number. If one assumes angle being dimensionless as defined in the SI, this might not be an issue, but then technically one could then just say "the measure of the angle" (without needing any qualification about units, since it is then (by definition) dimensionless, though one might want a footnote as many would find this confusing).

The SI is not the final word on angle being dimensionless: those who draft the SI are internally conflicted on the dimensionality of angle, some of whom have for many decades been proposing that angle be considered as being an independent dimension. This means that in an article such as this, we would do well to choose wording that is agnostic on the matter. Does anyone have a suggestion for such wording? 172.82.47.242 (talk) 22:30, 13 September 2022 (UTC)


 * Your quotation comes from section . Apparently, you forget that "angle" refers to both a geometrical figure and its measure. So, your comparison with heights is useless as a height is a measure and nothing else. Also your "interpretation" in the first sentence is wrong: an angle (figure) cannot be "expressed" with a number, but it can measured with a number.
 * I have edited the sentence containing your quotation into "For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction." I do not see any way to use your comments to improve this sentence. D.Lazard (talk) 09:04, 14 September 2022 (UTC)


 * The word "angle" originally means a geometrical figure made of two rays emanating from a common vertex, but is also often used to describe the relative orientation between two geometrical objects or the specification/quantification of a rotation or reflection; it often also gets used as shorthand for the "angle measure" (in radians, degrees, or the like). The measure of an angle in radians can be thought of as the natural logarithm of a rotation with the orientation stripped away; scaling between degrees and radians is like scaling between octaves and cents or between nepers and decibels. Even if you don't take the logarithm, rotations and relative orientations don’t really have a “unit” per se, but they do have a magnitude and orientation: they are best thought of as ratios of vectors (line-oriented magnitudes), bivectors (plane-oriented magnitudes), or the like. Anyhow, saying either "the measure of the angle in radians is π/2" or "the measure of the angle is π/2 radians" both seem clear enough.–jacobolus (t) 15:23, 14 September 2022 (UTC)
 * This last ("Saying either ...") tells be that my use of English in my first post is contrary to yours, and that "His height in metres is 1.83" makes sense to you, and not "His height in metres is 1.83 m". Fair enough – I'll not enter into a debate about the semantics of English phrasing, despite my reservations about choosing potentially ambiguous phrasing.
 * Indeed, bivectors provide an excellent way to encode rotation, but as with speed and velocity where one is referring respectively to a scalar and a vector quantity, and one should be clear which. In this article, it is clearly a scalar quantity that is meant, and there are useful nonsimple bivectors of rotation such as double rotations that do not translate to an angle per se, so I regard bivectors as clearly out of scope.  It seems to me that your use may seem so obvious to you that you cannot see how others might differently interpret the wording, and also that you have not been exposed to ongoing discussions around this in standardization bodies that could impact this article.  Anyhow, I'll leave this be.  172.82.47.242 (talk) 17:48, 14 September 2022 (UTC)
 * I think you are missing my point; as you suggest I was probably not clear enough in my description. Let me try again. You can represent a planar rotation or relative orientation between two coplanar vectors $$u$$ and $$v$$ of the same magnitude as the quotient $$R = v/u,$$ an object which multiplies by one vector to yield the other: $$Ru = (v/u)u = v.$$ This can be written as a "complex number" $$a + bI$$ where $$a^2 + b^2 = 1$$ and $$I$$ is a unit-magnitude bivector in the plane of the vectors. The logarithm of this rotation is the (bivector-valued) quantity $$\log R,$$ and if you strip the orientation out (picking arbitrarily between $$\pm I$$) you get an "angle measure" $$\theta = I^{-1}\log R$$ with units of radians. All of these quantities (except the vectors $$u$$ and $$v$$) are dimensionless. –jacobolus (t) 01:26, 15 September 2022 (UTC)
 * The prose sentence “His height in meters is 2.” is not idiomatic English, but if e.g. you asked “What is your height in meters?” then “2.” would be an entirely reasonable reply. Beyond that, the way the unit of “radians” is talked about/used doesn’t too closely match the usage of “meters” as a unit. –jacobolus (t) 01:35, 15 September 2022 (UTC)
 * I am familiar with geometric algebra and all the concepts that you refer to. Your argument that angle is dimensionless is no different from saying that the argument of the $sin$ function is dimensionless, and it is an angle, hence angles are to be considered dimensionless.  Which is a flawed argument, but this is not relevant here; my point is that it would be best to write this article in a way that is agnostic to this point since it is controversial even in the mainstream of metrology.
 * Any quantity can be considered to to have a dimensionless (numeric) part and a dimensional (unit) part, the latter itself being dimensionless for dimensionless quantities. Angles are no different from lengths or any other quantities in this respect.  We can consistently define angles to be dimensional, and to facilitate dimensional analysis, some people prefer to do so.  Mathematica does so, for example.  To be understandable to people working from either perspective, it would be good to use language that will be unambiguously understood about when we are referring to the quantity (including unit) and the dimensionless numeric part after removing (dividing by) the chosen unit.  I seem to have failed to communicate this above.  172.82.47.242 (talk) 13:45, 15 September 2022 (UTC)
 * These quantities are not dimensionless because they are the “numerical part” of a dimensional quantity. They are dimensionless because they are fundamentally ratios. The quantity “a right-angle rotation” is similar to the quantity “double” or the quantity “30%” in that respect. If you want you can put units on both sides of these ratios. Instead of “scale 1:100,000” on your map you can write “scale 1 cm:1 km” or whatever; in just the same way you can talk about a dimensionless angle measure multiplied by the radius (an arc length). You can similarly talk about the sine, secant, etc. of an angle as line segments or their lengths in whatever unit (as was done historically), or as an abstracted (dimensionless) ratio. –jacobolus (t) 15:18, 15 September 2022 (UTC)
 * Once you take the logarithm of a dimensionless quantity, you need to specify “units” again, in the form of the base of the logarithm. So for example you can say “10 doublings is approximately 3 orders of magnitude”. But these quantities are still “dimensionless” in the physical sense. You can apply 10 doublings to quantities measured in meters, seconds, bytes, or grams. –jacobolus (t) 15:23, 15 September 2022 (UTC)
 * Hoo-boy. We're clearly missing each other, and it is not because I am short on any conceptual or theoretical background here.  172.82.47.242 (talk) 15:32, 15 September 2022 (UTC)
 * If you read Molyneux's paper he presents a system where when you compute a logarithmic quantity it creates a logarithmic unit rather than dividing by a unit quantity. So for example one could take the (base 10) log of 10 and 100 meters, get 1 log-meter and 2 log-meters, and then subtract them to get 1 order of magnitude increase (i.e. 1 log-1). The log-meter is an "additive" unit and disappears on subtraction. So there are both dimensionless and dimension-ful logarithmic quantities.
 * Applying Molyneux's concept of logarithmic units to the definition of angle measure as the logarithm of a rotation is left as an exercise for the reader. Mathnerd314159 (talk) 18:28, 4 October 2022 (UTC)
 * Fair enough I guess (though I haven’t seen log-meters or the like before, and I’m not sure I buy that they have significant practical advantage). But an angle measure per se is still the logarithm of a quotient. If you wanted you could maybe take the logarithm of some position in a coordinate grid measured in meters, and end up with a (complex-valued) quantity measured in ln-meters. But the original units for the grid make no difference to the resulting imaginary component: the imaginary component of a particular point in the grid measured in ln-feet, ln-miles, and ln-meters is going to have the same numerical value. Taking the logarithm conveniently separates scale from rotation. –jacobolus (t) 18:51, 8 October 2022 (UTC)
 * FWIW, the log of a dimensional quantity is often used, wherever the decibel is used, with immense practical value. Treating it as "just a quotient" becomes somewhat strained in this context, since the "reference value" is arbitrary (just like a unit) and has indeed become auxiliary information that is attached to the unit.  —Quondum 12:36, 19 October 2022 (UTC)

clarifying basic definitions and scope
I think this article could use some significant rework to make definitions and scope more precise, and help especially non-technical readers. The word "angle" is used widely in many related but distinct senses, and it would be good to have a clear and well organized discussion of the relationships between these concepts. Angle can mean (at least):


 * a geometric figure formed by two line segments, rays, directed lines, circles, general curves, or the like (I would guess this is actually a significant minority use of all inbound wiki-links pointing at angle)
 * the relative orientation between two vectors, directed lines, or the like
 * at the most basic here "angle" means the (unsigned and unoriented) relation, a notion of the "distance" between an unordered pair of lines or orientations
 * it can mean the precisely specified rotation taking one object to another, oriented in a particular plane
 * it can sometimes allow for angles to be up to two full turns before returning to the identity state, as with unit quaternions; cf. plate trick
 * it can sometimes allow for a distinction between rotations which are any arbitrary number of full turns apart; this is especially relevant when an angle is used to measure e.g. the total rotation of a spinning physical object, the number of twists of a twist knot, or the like.
 * the angle of incidence between a line (or ray) and a surface; the dihedral angle between two planes or surfaces; in spherical geometry "lengths" are central angles and "angles" are dihedral angles.
 * a spatial coordinate in a circular coordinate system, representing a point on the circle; more generally one of the coordinates in a polar, spherical, etc. coordinate system, partially representing a point on a sphere, in space, or the like
 * the measure of an angle or other quantified representation of an angle such as a unit vector, unit-magnitude complex number, or slope.
 * the phase of a periodic function or the argument of a complex number, among similar concepts
 * an element ("number" or "point") in the periodic interval or circle group, abstracted from representing a specific relative orientation
 * the hyperbolic angle: analog of any of the above oriented in a pseudo-Euclidean plane (Minkowski or Lorentz plane); less commonly the "Galilean angle"(?) analog in a Galilean plane, perhaps usually just called slope. (In the hyperbolic plane, "angles" are (circular) dihedral angles while "lengths" are (hyperbolic) central angles.)
 * solid angle or other higher-dimensional analogs

These concepts are tricky and subtle, and I don't think we currently do an adequate job of organizing or representing them, especially for a non-technical or semi-technical audience. Does anyone know of any sources which explain these clearly? What do others think about the relative importance of these various concepts or ways of relating and describing them? Does anyone have any other recommended sources about the history/etc. of these concepts? –jacobolus (t) 05:26, 5 May 2023 (UTC)


 * I mean, there are a lot of bad/vague definitions of angle, but that doesn't mean WP has to be vague. And the way I see it your list of examples all points to the same definition. Considering general rigid transformations, they can be decomposed into rotations, translations, and reflections. Considering only rotations, they form a Lie group. A compact Lie group can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. A connected abelian Lie group is isomorphic to a product of copies of R (a vector) and the circle group S1 (an angle). The simple Lie groups are more complex and depend on the structure of the space. But focusing on the circle group, which is simply the complex numbers with magnitude 1, there is a natural labelling of its elements with angle measures, given by the argument or phase, and a natural generalization to the real line to avoid wrapping at $$2\pi$$.
 * As I see it, only a naive fool would forget the direction of a rotation! Counterclockwise and clockwise rotations are not the same at all. It is simply that angles have a "nice" structure where the inverse of a rotation has a negative angle. One can often be lazy and not worry about the sign, but this doesn't mean the concept of "unsigned" or "unoriented" angle has any real significance. Classical physics is time-reversible but this does not mean the past is the same as the future. Mathnerd314159 (talk) 16:32, 5 May 2023 (UTC)
 * People use the concept of unsigned and unoriented "angle" all over the place (with or without quantifying it), and there is nothing wrong with that. (As a simple example, consider the half-angle specifying the shape of a cone or the angular diameter of a ball as viewed from some particular point; these kinds of angles do not naturally come with a particular orientation, and giving them a sign is not really meaningful.) In a similar way mathematicians define whole disciplines in terms of a basic concept of unsigned and unoriented "distance" instead of only talking about oriented displacement vectors or the like. This doesn't make anyone defining such a concept a "lazy" "naïve fool", let's please stop with the exaggerated/insulting language, even when discussing people in the abstract. Wikipedia needs to describe the way these concepts are used in practice, not invent/canonize a single editor-preferred definition.
 * list of examples all points to the same definition – this is at best overly reductive. For example, in spherical trigonometry it is conventional to use a concept of "angles" at the separate vertices of a spherical triangle each of which is only given a 1-dimensional orientation intrinsic to the surface, but the dihedral angles those come from in the ambient 3-space are all oriented in different rotation planes; and a concept of "side lengths", each of which is given as a 1-dimensional quantity, even though the central angles those come from in the ambient 3-space are also all oriented in different rotation planes. We can make a list of identities relating these quantities without explicit reference to the 3-space orientations, but in a different context it would make sense to consider them as 3-dimensional bivector-valued (or unit-quaternion-valued) quantities, which would lead to a different set of relationships. (If you prefer we might consider the intrinsic 1-dimensional angles to also be bivector- or complex-valued, with bivectors of unspecified orientation.)
 * But perhaps more to the point, we need the definition here to be as legible as possible for a non-technical audience. Leading with anything about Lie groups is a non starter. –jacobolus (t) 17:39, 5 May 2023 (UTC)
 * I think any basic math article should discuss the most general formulation(s) of that concept. For example Derivative has a section devoted to generalizations of derivative. In the case of derivative, there are multiple generalizations, but here there is only one obvious concept (element of the circle group) that makes sense, so it is pretty much canonical, hence can take a more prominent role in the article.
 * I'm not saying the concept of unsigned and unoriented "angle" doesn't have a place, but I looked and there aren't any obvious references that argue against the signed definition. So the relationship I infer from the sources is signed as canonical and unsigned as a simplification, rather than the unsigned being canonical. I guess we could call the small angles "reference angles" (0-90 degrees) or "proper angles" (0-180).
 * I'm not sure what your point is with spherical trigonometry, the angles are still rotations and the sets of rotations around a point forms a circle group. The fact that the rotations form SO(3) and can be labelled with dihedral angles or 3D bivectors is interesting but not really relevant to defining angles.
 * As far as writing for a non-technical audience, WP:TECHNICAL is of course the guide. But notably there is WP:OVERSIMPLIFY: "Encyclopedia articles should not 'tell lies to children'". So dumbing down the definition of angle is not the right approach. I would say the lead paragraph should give a good definition in terms of the circle group and define general concepts, and then a second paragraph can then go on to describe common examples. Lie groups are relevant to defining angles in higher dimensions, but the basic notion of angle is defined in a plane, so the Lie groups can go in a section. Mathnerd314159 (talk) 18:33, 5 May 2023 (UTC)
 * Element of the circle group is too reductive (as you say, we should not WP:OVERSIMPLIFY). First, circle group explicitly describes such elements as complex numbers (not just as elements of an abstract group defined structurally with multiple possible representations), when the most popular representation of angles is as angle measures in a periodic interval. Second, this only covers angles in a particular plane, without allowing the possibility for angles to include orientation. Third, this does not allow for contexts where angle measure per se does not make sense, for example in referring to an angle in an affine space, or an angle in a space where the coordinates are elements of a finite field. Fourth, this does not allow for angles that can spin around many times, as seen in e.g. angular velocity. While we're at it it would be good (not necessarily in the lead but somewhere in the article) to describe the difference between absolute orientations vs. relative "angles" (quotients of orientations or if you like rotors transforming one rotation into another), cf. Baez (2009) "Torsors Made Easy". –jacobolus (t) 18:43, 5 May 2023 (UTC)
 * Where I will agree with you though is that I think we should refocus the lead definition here away from talking about a reference to particular parts of a geometric figure (though this sense of "angle" should be described early in the article) and toward a general concept of relative orientation. –jacobolus (t) 18:59, 5 May 2023 (UTC)
 * Yeah, the lead is the place to start. Maybe something like this:
 * An angle is formed when an object is rotated around a point. In Euclidean geometry, a ray is rotated around its initial point, called the vertex of the angle, and the starting and ending rays are called the sides of the angle. The measure of an angle denotes its magnitude and is a real number with some angular unit such as degrees or radians. The measure is in general any real number, but it may be bounded to [0,180°) or [0,360°) depending on context. Angles are often drawn as circular arcs, and the measure of a counterclockwise angle in radians is the ratio of the length of the circular arc to its radius.
 * Mathnerd314159 (talk) 20:28, 5 May 2023 (UTC)
 * I would instead make the fundamental definition something like "An angle is the relationship between two orientations or directed lines." But maybe we can find some sources (historical surveys, mathematical dictionaries, ...) with a clear definition. –jacobolus (t) 22:04, 5 May 2023 (UTC)
 * Unless you specify it as a rotation, you don't get angles beyond 360 degrees. Hence "relationship" is just too vague. Mathnerd314159 (talk) 14:43, 7 May 2023 (UTC)

"Angles" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=Angles&redirect=no Angles] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. —Lights and freedom (talk ~ contribs) 00:50, 23 May 2023 (UTC)

Euclidean Geometry only?
The lead sentence specifies euclidean geometry, but angles are also a thing in spherical geometry and hyperbolic geometry as well. Is there a common definition to all three that we can use in the lead sentence, and then briefly define the specifics for each geometry? It's a disservice to specify euclidean and then not discuss the other geometries as well, as doing so implies that only euclidean geometry has angles-- which is not true. Fieari (talk) 07:05, 23 May 2023 (UTC)
 * I went ahead and was WP:BOLD and did it. I welcome anyone to review my additions... I believe everything I said is accurate and lead-worthy, but I'd love a 2nd eye on it. Fieari (talk) 07:21, 25 May 2023 (UTC)
 * Your addition would make sense in a specific section named for example "In other geomeries". It is definitively too WP:TECHNICAL for the lead: most people interested in triangles do not know anything on non-Euclidean geometries. So, I'll revert your edit. D.Lazard (talk) 10:14, 25 May 2023 (UTC)

what about grad angle system? And forgot about circular angle?
I think we should add the following topics Yuthfghds (talk) 12:05, 4 June 2023 (UTC)


 * What do you mean by "circular angle"? Grads are discussed under . –jacobolus (t) 20:35, 4 June 2023 (UTC)
 * @Jacobolus Sorry my fault I didn't noticed the grad angle system.I am talking about degree measure in which 1°= 60' and 1'= 60''. Here we are only talking about linear angles not about circular degree measure.Thanks for pointing out my mistake, I am a new editor please avoid my faults.I still need feedback from your kind of editors.
 * Thank you
 * Yuthfghds Yuthfghds (talk) 05:49, 5 June 2023 (UTC)
 * I still do not understand what you are trying to say. What is the difference between "linear angles" vs. "circular degree measure"? This article does discuss degrees, minutes, and seconds. –jacobolus (t) 06:03, 5 June 2023 (UTC)
 * @Jacobolus
 * Linear Angles:
 * Linear angles are measured in degrees or radians and represent the amount of rotation or deviation between two lines or line segments.
 * They are used to measure angles in two-dimensional space, such as in geometry or trigonometry.
 * Linear angles have a fixed starting point (usually a reference line) and are measured in a straight line.
 * Linear angles range from 0° to 180° (or 0 to π radians) in a half-circle and extend up to 360° (or 2π radians) in a full circle.
 * Circular Angles:
 * Circular angles are measured in terms of arc length or fraction of a circle and are used to describe rotational positions or displacements.
 * They are commonly used in applications that involve circular or periodic motion, such as physics, engineering, or navigation.
 * Circular angles are based on the concept of a full circle, where one complete revolution is equal to 360 degrees or 2π radians.
 * Circular angles can exceed 360° or 2π radians, representing multiple revolutions or fractional parts of a revolution. Yuthfghds (talk) 06:17, 5 June 2023 (UTC)
 * "Linear angles" is not a term used in technical literature or curriculum that I have seen. Most of the results in a search for "linear angle" seem to be using it as a synonym for "straight angle" (i.e. a half turn, 180°).  Where did you get these names? Do you have a source? I would call this instead something like "angular distance" or "angular separation" (or just "angle"). I agree that this article should do a better job about talking about different concepts of angle, and I have started gathering sources for hopefully an eventual rewrite/rework. –jacobolus (t) 06:45, 5 June 2023 (UTC)
 * @Jacobolus Sir, here are some references for linear and circular angles:- help you.
 * Linear Angle:
 * Elementary Geometry for College Students by Daniel C. Alexander and Geralyn M. Koeberlein.
 * Chapter 1: Basic Geometry Concepts.
 * Chapter 2: Reasoning and Proof.
 * Geometry: A Comprehensive Course by Dan Pedoe.
 * Chapter 2: Angles.
 * Circular Angle:
 * Trigonometry by James Stewart, Lothar Redlin, and Saleem Watson.
 * Chapter 1: Trigonometric Functions.
 * Trigonometry by I.M. Gelfand and Mark Saul.
 * Chapter 1: Angles.
 * Elementary Trigonometry by H. S. Hall and S. R. Knight.
 * Chapter 1: Trigonometrical Ratios of Acute Angles.
 * sir if you still have some queries please question me.Thanks for feedback.
 * Yuthfghds Yuthfghds (talk) 07:25, 5 June 2023 (UTC)

Unless I am doing something wrong with my text searches at the internet archive, the term "linear angle" does not appear in either of the books you listed by Alexander or Pedoe. –jacobolus (t) 11:13, 5 June 2023 (UTC)


 * @Jacobolus Please recheck,if you still don't find it I can't do anything.Because I don't have those books that I mentioned you.Because I red those books few years ago.And I think linear angle is informal term mentioned by the author.And I argued here so it's my fault but still we can add these topics in this article.If I tell you the definition of linear angle and circular angle it would be like this:-
 * linear angle- Angle measured between two rays in degrees.
 * circular angle- Angle measured between two rays also in degrees but including minutes and hours. Technically angles that are placed inside a unit circle with 3 divisions namely degrees, minutes and hours.
 * I am glad that you understand my perspective and debate over this topic. Yuthfghds (talk) 13:03, 5 June 2023 (UTC)


 * The angle between two rays does not depend on whether or not a circle is drawn around their vertex! — When you say "including minutes and hours", are you referring to right ascension (part of the coordinates of a star in the sky) which is at least sometimes given in hours? — The subdivisions of a degree are called minutes and seconds. As far as I know, hours and degrees are never used together, partly because of the confusion between two kinds of "minutes". —Tamfang (talk) 21:01, 7 July 2023 (UTC)
 * @Tamfang You are absolutely right.These are just "informal" terms to distinguish between trigonometric angles and geometric angles. Yuthfghds (talk) 05:25, 9 July 2023 (UTC)
 * Why do they need distinguishing? Is there ever a circumstance in which mistaking a trigonometric angle for a geometric angle (or vice versa) or a linear angle for a circular angle (or vice versa) (are these the same distinction?) would lead to wrong results? —Tamfang (talk) 20:08, 5 August 2023 (UTC)


 * As I mentioned above linear angle is an informal term referred to angles that are measured in only degrees like in protector when we measure angle we say 12.5° or 45.2° just because we can't expand it but in circular angles we expand this system.linear angles are helpful in geometry but circular angles are helpful in trigo Yuthfghds (talk) 13:12, 5 June 2023 (UTC)


 * well the references that I mentioned you are generated by Chatgpt.But I red the book by padeo don't know linear angle is in that book or not but I red somewhere about linear angle in a random book but when Ai generated that book I thought it cover that topic sorry for the inconvenience that you suffered but still together we can make this article near excellence. Yuthfghds (talk) 14:14, 5 June 2023 (UTC)


 * Please don't ever ask ChatGPT for help writing Wikipedia, and be careful using it for any other purpose. It will confidently make up total nonsense. ChatGPT is a statistical language model, not a reference librarian. –jacobolus (t) 15:36, 5 June 2023 (UTC)


 * @Jacobolus You are absolutely right ,my bad. Yuthfghds (talk) 15:47, 5 June 2023 (UTC)

Negative angle definition
The following was removed: "Although the definition of the measurement of an angle does not support the concept of a negative angle," as unhelpful. Of course a definition can be contrived. Essentially one refers to area of a circular sector, using signed area according to the area being above or below the horizontal line of symmetry. This definition is for angles fixed in a standard position. Reference to area for definition is the historic basis of hyperbolic angle, so angle reference to area is one of the unifying theories in mathematics. Rgdboer (talk) 23:03, 9 November 2023 (UTC)


 * This article needs to be clarified and expanded to more clearly describe and discuss the several different concepts of "angle" which are in common use. This is something I plan to do eventually; I've been occasionally gathering sources at User:Jacobolus/Angle but am not yet ready to write a solid survey. –jacobolus (t) 00:45, 10 November 2023 (UTC)

Thank you for those sources on directed angle. The issue of angle definition came to a head in 1893 in Chicago when Felix Klein shot down Alexander Macfarlane’s paper on the topic which was presented to the mathematical congress held in connection with the Columbia Exposition. The paper was included in Papers in Space Analysis (1894) as the Proceedings of the congress only noted the title. The notion of an area-based definition referring to the sector of a circle was included in The Elements of Plane Trigonometry (1892) by R. Levitt & C. Davison (page 158). Robert Baldwin Hayward noted the complete analogy between the circular and hyperbolic cases when area of sectors are used to define the circular and hyperbolic angles. — Rgdboer (talk) 01:19, 12 November 2023 (UTC)


 * Of course, both circular and hyperbolic angle measure can be defined either via area or via arc length, with the latter using a Lorentzian (pseudo-Euclidean) concept of distance in the plane. –jacobolus (t) 02:25, 12 November 2023 (UTC)