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Turn is period?[edit]

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? Dbfirs 07:29, 15 September 2013 (UTC)Reply[reply]

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)Reply[reply]
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. Dbfirs 06:25, 18 September 2013 (UTC)Reply[reply]

Transversal and polygons both have interior angles and exterior angles[edit]

Comments at the article edit: "I don't see what the association is for polygons. No mention at the linked article." See Angle#Polygon_related_angles. 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)Reply[reply]

Small article merger[edit]

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 (talkcontribs) 21:25, 24 September 2013 (UTC)Reply[reply]

   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
--Jerzyt 07:16, 21 July 2014 (UTC)Reply[reply]
Note the date stamps: I did the merges a few months after the comments by Reddwarf2956. RockMagnetist (talk) 15:41, 21 July 2014 (UTC)Reply[reply]
   That made no sense to me on my first several readings, and i responded at User_talk:RockMagnetist#Talk:Angle.23Small_article_merger, 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.
--Jerzyt 10:14, 25 February 2017 (UTC)Reply[reply]


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:


and radians as:

(s/c)×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)Reply[reply]
"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)Reply[reply]
   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.
--Jerzyt 16:30, 20 July 2014 (UTC)Reply[reply]
   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.)
--Jerzyt 16:30, 20 July 2014 (UTC)Reply[reply]

talk page destruction![edit]

Who eviscerated this talk page? Yesterday it had 36 sections and today it has 4?!! Skylarkmichelle (talk) 22:43, 20 December 2013 (UTC)Reply[reply]

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)Reply[reply]

Proposed merge[edit]

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)Reply[reply]

Support, with sources added. Otherwise delete. - DVdm (talk) 19:21, 4 October 2014 (UTC)Reply[reply]
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)Reply[reply]
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 Wikipedia: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)Reply[reply]

Supplementary & Complementary[edit]

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.) Dbfirs 07:05, 14 July 2016 (UTC)Reply[reply]

It's all in the reflexes[edit]

What about reflex angles? — Preceding unsigned comment added by (talk) 17:58, 24 November 2016 (UTC)Reply[reply]

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)Reply[reply]

Merger of Angular unit into Angle#Units[edit]

they overlap widely. fgnievinski (talk) 05:09, 7 May 2021 (UTC)Reply[reply]

Oppose: they are in appropriate Summary style. Klbrain (talk) 20:00, 22 May 2021 (UTC)Reply[reply]
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#Units. It does not merit a separate page in its current form. —Quondum 20:15, 22 May 2021 (UTC)Reply[reply]
Support per nom. In reply to Klbrain, there is not enough unique and encyclopedic material in Angular unit to merit invoking WP:SUMMARY. — Cheers, Steelpillow (Talk) 17:40, 9 July 2021 (UTC)Reply[reply]
All fair points, which I'm happy to concede. No objections to a merge from me. Klbrain (talk) 22:57, 9 July 2021 (UTC)Reply[reply]
  checkY Merger complete. Klbrain (talk) 15:40, 1 August 2021 (UTC)Reply[reply]


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)Reply[reply]

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)Reply[reply]
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)Reply[reply]
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)Reply[reply]
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)Reply[reply]

"Ángulo" listed at Redirects for discussion[edit]

Information.svg An editor has identified a potential problem with the redirect Ángulo and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 February 14#Ángulo 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)Reply[reply]

"Ângulo" listed at Redirects for discussion[edit]

Information.svg An editor has identified a potential problem with the redirect Ângulo and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 February 14#Ângulo 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)Reply[reply]

Proposed merge of Angle of rotation into Angle[edit]

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.
The result of this discussion was…merge to Angle .

Joyous! | Talk 22:55, 19 February 2023 (UTC)Reply[reply]

"Angle of rotation" is already covered in Angle. fgnievinski (talk) 03:43, 29 March 2022 (UTC)Reply[reply]

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)Reply[reply]
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)Reply[reply]
Support the alternative target, Rotation. It would be well placed there, perhaps before Rotation#Axis of 2 dimensional rotations which explicitly uses the angle of rotation without describing it in one dimension. Klbrain (talk) 10:08, 2 January 2023 (UTC)Reply[reply]
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)Reply[reply]
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)Reply[reply]
  • 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.
The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Looks like everyone is in solid agreement. Courtesy pinging @Fgnievinski, Mathnerd314159, Quondum, Klbrain, and Jacobolus: . 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)Reply[reply]

  checkY Merger complete.Quondum 01:41, 23 February 2023 (UTC)Reply[reply]


Like what is explanation of angles (talk) 08:41, 6 May 2022 (UTC)Reply[reply]

Dimensional subtlety[edit]

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? (talk) 22:30, 13 September 2022 (UTC)Reply[reply]

Your quotation comes from section § Alternative ways of measuring an angle. 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)Reply[reply]
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)Reply[reply]
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. (talk) 17:48, 14 September 2022 (UTC)Reply[reply]
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 and of the same magnitude as the quotient an object which multiplies by one vector to yield the other: This can be written as a "complex number" where and is a unit-magnitude bivector in the plane of the vectors. The logarithm of this rotation is the (bivector-valued) quantity and if you strip the orientation out (picking arbitrarily between ) you get an "angle measure" with units of radians. All of these quantities (except the vectors and ) are dimensionless. –jacobolus (t) 01:26, 15 September 2022 (UTC)Reply[reply]
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)Reply[reply]
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. (talk) 13:45, 15 September 2022 (UTC)Reply[reply]
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)Reply[reply]
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)Reply[reply]
Hoo-boy. We're clearly missing each other, and it is not because I am short on any conceptual or theoretical background here. (talk) 15:32, 15 September 2022 (UTC)Reply[reply]
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)Reply[reply]
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)Reply[reply]
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)Reply[reply]