Talk:Angle

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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)[reply]

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)[reply]

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)[reply]

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)[reply]

"Anggulo" listed at Redirects for discussion

The redirect Anggulo 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 Wikipedia:Redirects for discussion/Log/2024 August 23 § Anggulo until a consensus is reached. 1234qwer 1234qwer 4 23:09, 23 August 2024 (UTC)[reply]

"Ajacent angels" listed at Redirects for discussion

The redirect Ajacent angels 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 Wikipedia:Redirects for discussion/Log/2024 August 23 § Ajacent angels until a consensus is reached. 1234qwer 1234qwer 4 23:10, 23 August 2024 (UTC)[reply]

plane angle

Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection. Angles are also formed by the intersection of two planes; these are called dihedral angles. In any case, the resulting angle is also known as plane angle, as it lies in a plane (spanned by the two rays or perpendicular to the line of plane-plane intersection).

If this last sentence is to distinguish from solid angle, I think that need not be in the lede. —Tamfang (talk) 03:44, 9 January 2025 (UTC)[reply]

Previously it implied dihedral angles were not plane angles. Now in a third iteration, plane angle is just given as a synonym for angle. fgnievinski (talk) 19:21, 10 January 2025 (UTC)[reply]

Suggestions for page

There's a lot of good info on this page, but I have some suggestions which I will start making. Please let me know thoughts, objections or alternatives.

Lead - Should the information about tangents and dihedral angles be in the lead? This seems fairly technical to be presented first up and might be better left as part of section on types of angles.

Units of angles - This needs to be (briefly) introduced early in the article as all sections make reference to degrees, radians tuns and so on, before these are defined or explained. More detailed information about this topic can come further down, but readers need to know broadly that there are different units of measurement and what they are before these are used in the article.

Identifying angles - suggest that this section could be better titled as Notation.

History and etymology - is there any WP convention as to where this section should go? To me, it's something I would look at after getting a handle on the basic concepts, so would prefer to have more towards the end, but interested to hear other viewpoints. TheGrifter80 (talk) 12:37, 21 May 2025 (UTC)[reply]

I've wanted for a long time to substantially rewrite this whole article; I think it's quite mediocre and confusing. We should be explaining clearly that the word "angle" is used to mean several distinct but related things. I'm not sure the best way we can keep it simple while also being clear.

Some of the possible meanings of "angle" include:

a vertex of a polygon
a specific pair of rays or line segments meeting at a vertex
a specific pair of curve segments meeting at a vertex, or the pair of tangent rays where the curves intersect
the class of all pairs of rays with a common vertex which can be superimposed by isometries
the portion of the plane lying between two rays which meet at a common vertex
the class of all such portions of the plane which can be superimposed, perhaps represented by their relative quantity, which might be thought of as the constant area ratio between a "pie slice" and a full disk
the unsigned "measure" of an angle, proportional to the length of a circular arc of standard radius between two rays or between the tangent rays to two curves
a specific rotation transformation
the class of all rotations which rotate the plane by the same amount in the same direction (these can be compared e.g. by looking at a "before" ray for each rotation with vertex at the center of one rotation and its "after" image under the rotation, and then performing a translation and rotation from one "before" ray onto the other and performing the same transformation on the "after" ray, and seeing whether it has been superimposed on the other "after" ray)
the signed "measure" of a rotation angle, proportional to a directed arclength of a circular arc.
the class of all rotations which rotate a higher-dimensional Euclidean space by the same amount in the same orientation
the oriented "measure" of a rotation angle in space, a quantity which can be an arbitrary bivector in the space

All of these can also be generalized to pseudo-Euclidean space.

To be clear, the list above is too long and confusing to be included in our initial description, but we should give some sense of the multiplicity of meanings and their history rather than pretending that there's just one definition. –jacobolus (t) 18:17, 21 May 2025 (UTC)[reply]

That's a great point and you've put it very well. As you say the challenge is to communicate that without overwhelming the reader.

Perhaps we start off giving a more plain language description first, highlighting that the word angle is used to refer to many different "things" in geometry, that generally relate to the intersection of straight lines.

Then we could introduce a few of the more fundamental definitions of the ones you've identified, which to me would be:

1. vertices of a polygon (probably the most commonly encountered example for the average reader)

2. Intersection of two rays (seems to be the standard definition in geometry)

3. Measure of angle. I think it's important to stress up front that there is no seperate word for the size of an angle. Eg while a line (object) has a length which is a quantity (number plus unit) an "angle" (object) has an "angle" which is a quantity (number plus unit). TheGrifter80 (talk) 01:07, 22 May 2025 (UTC)[reply]

There is a separate term for the size of an angle: "angle measure". But often this is just also called "angle", and the two concepts are routinely conflated. "standard definition in geometry" – I don't think this is really so standard; in my experience the word "angle" more often refers to either the equivalence class of such pairs of rays or some quantitative representation of it. It's also worth discussing some of the other representations of an angle, for example a unit-magnitude "complex number" (scalar number part + bivector part; in 3-dimensional space, an oriented angle can be represented by a versor), or the half-tangent (trigonometric tangent of half the angle), a real number ranging from ⁠

-\infty

⁠ representing a clockwise half-turn through ⁠

+\infty

⁠ representing an anticlockwise half-turn. –jacobolus (t) 03:05, 22 May 2025 (UTC)[reply]

New section: Fundamentals

I've created a new section called Fundamentals. The purpose is to give the reader a broad overview of some of the fundamental concepts used to discuss and understand angles without going into full detail of each one. But I think there is too much info required to cover this in the lead.

One challenge with this type of foundational topic is that all of the concepts are interlinked. For example the section on Types uses degrees, radians and turns to describe them so it seems reasonable to describe what these units are first. But for the average reader it's probably not useful to go into the full detail of how radians are defined before introducing more basic ideas like classification/names of different types of angles. Please let me know thoughts. TheGrifter80 (talk) 01:21, 27 May 2025 (UTC)[reply]

Recent revert

@ZergTwo: Regarding the recent revert, you wrote "Non-neutral writing, unsourced content." The sources are all in the Radian article, they are quite substantial so I did not have time to add them. As far as neutrality, can you explain what is not neutral? That section in the Radian article has been the subject of significant debate and my summary of it is accurate as far as I can tell. Mathnerd314159 (talk) 22:30, 28 May 2025 (UTC)[reply]

I would have partially reverted your edit if you did add the sources. Also, your expansion of Angle#Dimensional analysis contains subjective expressions like tricky to explain, feels inconsistent, and many others. I suggest being familiar with Wikipedia:Neutral point of view#How to write neutrally and Wikipedia:Manual of Style/Words to watch to avoid similar issues. ZergTwo (talk) 23:35, 1 June 2025 (UTC)[reply]

The direct quotes are "a perennial problem in the teaching of mechanics" and "pedagogically unsatisfying". They are unopposed so they can be presented as fact - "how to write neutrally" applies only to biased statements of opinion. To paraphrase WP:SUBJECTIVE, "it is permissible to use a [subjective expression] when that opinion is widespread and potentially informative or of interest to readers." Regarding words to watch, "tricky to explain" and "feels inconsistent" fall under none of the categories. Mathnerd314159 (talk) 05:11, 2 June 2025 (UTC)[reply]

"Circular measure"

The article has the sentence Broadly there are two approaches to measuring angles: relative to a reference angle angle (such as a right angle); and circular measurement.

This statement seems problematic to me:

The term seems to be "circular measure" (from which this article presumably derives "circular measurement").
There seem to be few authoritative sources that define this term. Searches turn up low-quality online pedagogy and dictionary entries. It seems to be more historical than modern.
The only clear definition of "circular measure" that seems to make sense is the ratio of the subtended arc length to radius for an angle; note that this quantity is inherently dimensionless. This definition stands in contrast to the size an angle with respect to a reference angle, which need not be dimensionless, and essentially needs a unit. However, the article does not clarify this distinction.

The SI has unfortunately clouded this issue by conflating the two quantities defined and has done so for generations, undercutting the ability of people to think about this issue, leaving an incredibly small number of people who can reason clearly about this. One only needs to see the ongoing decades-long (or centuries-long?) disagreements between the top metrologists in the world to understand how true this is.

I am unsure as to the best approach. I see three options:

Clearly define the two approaches and the distinction between in the quantities generated in the two (but which will likely be difficult to source)
Take the approach of SI and mathematics and pretend that they are the same, perforce defining the dimension of the measure of an angle as being dimensionless
Word the article as agnostic to this: choose words that apply to all interpretations, and highlight when the SI approach is specifically being referred to.

I prefer the first but it has the obvious problem of sourcing; I find the second approach abhorrent because it perpetuates a clearly problematic position; so in the past I have kept the article in line with the third approach, which is at least intuitive wiyhout being controversial.

Should we just delete the quoted statement, or is there an uncontroversial way to elaborate on the distinction between the two approaches and the quantities that they produce? —Quondum 14:39, 10 August 2025 (UTC)[reply]

I agree with you that the section is not quite yet clear and precise enough. But to me it is fair to say that there are two methods/systems for measuring angles. One gives degree measure, the other gives radians. They aren't just arbitrarily different units (and as you say is the radian actually a unit or not?), in the way metres and feet are different measures of length.

I agree with you that first option you suggest is best. What do you think needs further clarification here? Perhaps rather than using the terms "circular measure/measurement" would it better to "measurement using a circle" or "radian measure"? TheGrifter80 (talk) 00:47, 11 August 2025 (UTC)[reply]

Here are some sources I've been gathering for a while:

Concepts of angle:

Kontorovich, Igor'; Zazkis, Rina (2016). "Turn vs. shape: teachers cope with incompatible perspectives on angle". Educational Studies in Mathematics. 93 (2): 223–243. JSTOR 24830689.
Freudenthal, Hans (1983). Didactical Phenomenology of Mathematical Structures. Dordrecht, Holland: D. Reidel.

Directed angle:

Chasles, Michel (1880) [1st ed. 1852]. Traité de géométrie supérieure (2nd ed.). Paris: Gauthier-Villars.
Court, Nathan Altshiller (1952) [1st ed. 1925]. College Geometry (2nd ed.). Barnes & Noble.
Chick, J.M. (2017). "101.39 Another angle on angle?". Mathematical Gazette. 101 (552): 502–508. JSTOR 26538939.
Blaga, Cristina; Blaga, Paul A. (2018). "Directed Angles" (PDF). Didactica Mathematica. 36: 25–40.

Sum of angles in a triangle:

Burke, Maurice (1992). "Circles Revisited". Mathematics Teacher. 85 (7): 573–577. JSTOR 27967774.

Angle measure:

Jones, Phillip S. (1953). "Angular Measure – Enough of Its History to Improve Its Teaching". Historically Speaking. Mathematics Teacher. 46 (6): 419–426. JSTOR 27954356.

Phase:

Lawson, Jeffrey; Rave, Matthew (2016). "Spacewalks and Amusement Rides: Illustrations of Geometric Phase". Mathematics Magazine. 89 (2): 105–121. doi:10.4169/math.mag.89.2.105.

Generalizations and advanced concepts:

Gottlieb, Daniel Henry (1996). "All the Way with Gauss-Bonnet and the Sociology of Mathematics". American Mathematical Monthly. 103 (6): 457–469. JSTOR 2974712.

I'm not quite at the point I'm ready to tackle a significant rewrite of this article yet though. –jacobolus (t) 02:22, 11 August 2025 (UTC)[reply]

My thought would be to just discuss that the most common ways of measuring an angle are degrees and radians, and then have a section for degree and a section for radians, linking to the appropriate articles with hatnotes. We can also make the units table shorter by moving the information about degrees and radians out of that table. As far as a classification I would say the statement is wrong because for example you can measure angles by their slope or tangent, as in the related quantity section, and such a measurement uses neither a circular measure nor a reference angle. Mathnerd314159 (talk) 04:24, 11 August 2025 (UTC)[reply]

The article eventually should have a section about alternate ways of representing angles, including by complex coordinates (sine and cosine) or equivalently a 2x2 matrix, the half-tangent, the square of the sine (identifies the angle between un-oriented lines, rather than rays, but for lines whose slope is an element of an arbitrary field is also an element of the same field), and perhaps others. The half-tangent is particularly important in many practical applications, especially in subjects like the kinematics of linkages, as found in e.g. robotics. –jacobolus (t) 05:01, 11 August 2025 (UTC)[reply]

That is a fair suggestion to use degrees and radians. My view would be that some discussion of these two measures on this page would be useful for the reader to understand and contrast the two different approaches. TheGrifter80 (talk) 06:43, 11 August 2025 (UTC)[reply]

jacobolus, "Turn vs. shape: teachers cope with incompatible perspectives on angle" talks about a second shortcoming of the article: the confusion that results from using the same term "angle" to refer to both the figure and a measure of the amount of rotation needed to make the one ray incident with the other. This would be easily addressed by choosing separate terms and using them consistently. I don't think that this is conceptually challenging, though. I have not looked through the content of your references, but nothing there strikes me as addressing the issue of incompatible measures of angle.

Mathnerd314159, you don't seem to directly address the core issue: how to make clear that "different ways" of obtaining the size of an angle leads to incommensurate quantities. Avoiding the issue like this is embodied in my third option. It would be like saying that we can measure the size of a circle as the distance from its centre to its circumference, or as the distance around its circumference, but such a description fails to alert the reader that these two measures of size are incompatible. In the case of "size of an angle", this leads to widespread confusion, because the difference between the derived quantities is more subtle. I agree that there are more ways than two to quantify the size of an angle. The point of that statement really should be to say that circular measure and in terms of a reference rotation amount are incommensurate, not to give the number of ways of classifying them. —Quondum 13:41, 11 August 2025 (UTC)[reply]

All linear angular measures — degrees, radians, gradians, etc. — are connected by fixed proportionality constants. In metrological terms, that means they are fully commensurable: you can convert between any of them by multiplying by a constant, so "incommensurate" isn't the right term here. The question of whether the radian is treated as dimensionless or as having its own dimension is a matter of convention and notation, not a fundamental difference in the underlying quantity—it's still a linear angular measure either way. For this section, the key point is that multiple angular measures exist and are directly interconvertible; the more detailed discussion of dimensions belongs in the dimensional analysis section and at Radian. Mathnerd314159 (talk) 15:57, 11 August 2025 (UTC)[reply]

You may be right in terms of my use of the term, but multiplying constants are non-trivial here. The radius and circumference are exactly proportional and are both a linear measure of the size of the circle either way, but are you going to argue that there is not a fundamental difference in the underlying quantity? As another example, it makes sense to regard angular frequency as a different quantity from frequency, otherwise mathematical expressions involving these quantities become inconsistent. And that there confusion stemming from this is indisputable. I even had an argument with an editor who vociferously argued that 1 Hz = 2π rad/s, so much so that I ended up abandoning editing the article in question. Writing something off as "a matter of convention" does not address this. —Quondum 16:21, 11 August 2025 (UTC)[reply]

In my understanding "Hertz" just means "cycles per second", and doesn't inherently imply anything proportional to [physical or abstracted] arclength. –jacobolus (t) 18:31, 11 August 2025 (UTC)[reply]

Exactly: the SI definition of Hz is essentially one event count per second, with "event count" being the dimensionless number 1. My point is that in the current system, few people can reason clearly about it. —Quondum 19:33, 11 August 2025 (UTC)[reply]

Well, for example with the circle, whether you measure the radius, the circumference, the diameter, or any other aspect such as curvature, you're still describing the same physical feature — its size — and you can convert between them directly. They're different measurements in the sense of "different aspects you can measure", but they all refer back to the same underlying feature of the circle. It's similar with the volume of a box: you could measure its sides and multiply them, or submerge it in water and measure the displaced volume. The methods differ, but the result describes the same property — the box's volume. In this case, the volume is a physical quantity so these measurements will give close to the same result when you take into account the units used, but I don't think that is too relevant here. Now with angles, you can measure them with a protractor, by the arc-length method, or by calculating a slope. These approaches yield different numbers and units, but they all quantify the same thing — the size of the angle. Conventions about units, dimensions, and the quantity measured do matter for clarity, and the frequency vs. angular frequency distinction is a good example, but they don't change the fact that degrees, radians, and other angular measures are directly convertible and do indeed fundamentally measure "the same thing". Mathnerd314159 (talk) 20:56, 11 August 2025 (UTC)[reply]

"they all quantify the same thing — the size of the angle"

This is oversimplified. Even if we limit ourselves to one-dimensional angles, there are multiple different things we might want to describe: (1) the relative orientation of two directed lines (or vectors or rays) in n-dimensional space, (2) the signed extent of that relative orientation, starting from a predefined oriented plane in which the two rays lie, (3) the extent of that relative orientation, irrespective of the plane in which the lines lie, (4) one or another motion taking one of the oriented lines into the other (the required motion will be different if we require that a particular point on the first line must match a particular point on the second line after the motion), (4) the relative orientation of two undirected lines, (5) the portion of the field of view from a given center point subtended by a given other object, ...

Now for each of these possible kinds of things we want to quantify, there are multiple possible quantitative representations, some of which are insufficient or excessive for the specific purpose. The various representations which precisely match the purpose compose in different (but isomorphic) ways, but not all of the representations have the same expressive power. –jacobolus (t) 21:29, 11 August 2025 (UTC)[reply]

Mathnerd314159: I think you are still missing Quondum's point, which from what I can tell is that an angle doesn't inevitably need to be represented using angle measure. The way I personally like to think of things, an angle (rotation, relative orientation, or the like) should be taken to compose multiplicatively in its canonical form, and is usually a (bivector + scalar)-valued quantity with the bivector oriented in the plane of rotation. The arclength-proportional "angle measure" is a logarithm of the multiplicative angle (typically divided by the unit bivector of the same direction, which strips orientation out leaving a scalar quantity), and like other logarithms, converts multiplication to addition, so that angle measures compose additively. When angle measures are used to represent e.g. time in a periodic signal or the domain of an abstracted periodic function, then the implicit orientation is likewise an abstract/"imaginary" quantity rather than a physical orientation. –jacobolus (t) 18:35, 11 August 2025 (UTC)[reply]

I think that was Mathnerd314159's point, with which I agree. My point is more that if we avoid conflating related quantities in our minds as described, even when they might seem to be the same thing, reasoning and mathematical expressions start working together much more intuitively. And to add to your point of a bivector representing angle-of-rotation quantities, it would be nice if the concept was better-known, and maybe it will get there eventually. —Quondum 19:33, 11 August 2025 (UTC)[reply]

I would say the bivector is "more than an angle" — typically one would say angles are preserved under similarity, but with the bivector, rotating the plane will also rotate the plane of the angle and change its values. I'll agree the bivector stuff is interesting but based on the textbooks and WP:ONEDOWN I would say this page should be aimed at middle school and getting into anything beyond 2D geometry would be too advanced. I would say we definitely need a page for more advanced treatments of angles - maybe the philosophy and history stuff could be expanded into an article discussing the development from the Greeks to quaternions and bivectors and so on, covering 3D and higher D discussion, and all those archaic units that got deleted (which do have reliable sources, they are just too obscure for a middle-school level page). Mathnerd314159 (talk) 21:11, 11 August 2025 (UTC)[reply]

Quite the contrary, this page should endeavor to cover all relevant aspects (at least in summary), starting from basic ones and working toward trickier ones. Not every reader needs to fully understand every word of the page. We should discuss and contextualize undirected and directed angles, oriented angles in 3+-dimensional contexts, angles in pseudo-Euclidean space and hyperbolic rotations, dihedral angles, orientations of planes in 3+-dimensional space, solid angles, angles between curves in arbitrary (pseudo-)Riemannian manifolds, angle-preserving (conformal) geometric transformations, the use of "angle" in abstracted/non-geometrical contexts such as periodic signals, and so on. –jacobolus (t) 21:31, 11 August 2025 (UTC)[reply]

I guess it is possible to explain such topics at a middle school level. But the lead sentence since 2007 has been "In Euclidean geometry, ...". It seems like it would be hard to change this incrementally. I still think creating a new page is the way to go - it could replace this page, perhaps, but only once it achieves a decent quality. Mathnerd314159 (talk) 03:57, 12 August 2025 (UTC)[reply]

As for weird units for angle measures: Our discussion at this article shouldn't separately list arcseconds or milliradians (and I'm not sold on grads or binary degrees either). At this article a few paragraphs would be better than a table, or if there is a table its columns should also be pared down. A whole separate article about Units of angle measure would be a good place for further detail, including a table with whatever obscure (sourced) units someone wants to throw in. –jacobolus (t) 21:59, 11 August 2025 (UTC)[reply]

What is the "measure" of angle?

The above discussion raised some good points about angle measure. Would it be useful to discuss in the article some different perspective of exactly "what" the measure of angle is (prior to discussion of "how" to measure an angle)? As far as I can see there are a few different approaches in textbooks and literature:

Left undefined
Smallest amount of rotation about the vertex of one ray to the other
"Amount" of opening between the sides
Length ratios: arc length to radius (SI approach) or arc length to circumference (degrees)

Would some discussion of this be useful or is it too confusing for an article at this level? TheGrifter80 (talk) 00:23, 12 August 2025 (UTC)[reply]

You also forgot an area ratio. We might at some point also mention the integral of curvature with respect to arclength between two points on a curve, though probably better to keep that for later in the article. –jacobolus (t) 00:47, 12 August 2025 (UTC)[reply]

Does integral of curvature make sense in other than a plane? Since angles of rotations do not "add" (adding two bivectors is commutative, but the two rotations that they induce do not commute other than in a plane); this suggests that integrating of a "curvature bivector" will not work either. —Quondum 01:17, 12 August 2025 (UTC)[reply]

I think it makes sense on an arbitrary Riemannian manifold using a scalar concept of curvature. –jacobolus (t) 02:09, 12 August 2025 (UTC)[reply]

I've thought about it enough to be doubt that, in other than a plane, anyone would give the name "angle" to the integrated scalar curvature along a curve. In Euclidean space, for instance, it would give at best (half of) the angular aperture of the cone that bounds the final direction. —Quondum 12:38, 12 August 2025 (UTC)[reply]

Oh, I meant a curve on some 2-dimensional surface; not sure about space curves. Anyway, I didn't mean to derail this topic. –jacobolus (t) 15:31, 12 August 2025 (UTC)[reply]

Ah – yes. There is a higher-dimensional analogue in higher-dimensional manifolds involving the concept of parallel transport, but it is not of a scalar, it is not true integration, and it probably also requires a given affine connection, such as the Levi-Civita connection. —Quondum 16:04, 12 August 2025 (UTC)[reply]

Hardy

According to G. H. Hardy in his Course of Pure Mathematics, the fatal defect in definition of angle as the length of an arc is the difficulty in defining the length of an arc.(page 316) Instead he shows that the angle of a line of slope m is $\arctan m=\int _{0}^{m}{\frac {dt}{1+t^{2}}}$ . This result is obtained (page 317) with algebra and integration, referring to a figure on page 316. Later in the book in a section on analytical theory of circular functions he summarizes: "Either with any arc of a circle is associated a number which we call its length, or that with any sector of a circle is associated with a number which we call its area. These demands are alternative, and when either of them has been satisfied our trigonometry will rest on a secure foundation. It is usual to adopt the first alternative, and to base trigonometry on the notion of length; but Ch. VII contains an accurate discussion of areas and not of lengths, so that we were naturally led to prefer the second alternative."(page 433) Elsewhere (b:Geometry/Unified Angles) the full theory of angle, including the hyperbolic variety, is developed after the area fashion preferred by Hardy. Rgdboer (talk) 21:18, 12 August 2025 (UTC)[reply]

Providing different approaches to defining angles is attractive: I can imagine people who want to start an exploration on the topic needing more than the classroom definitions. Length of arc does need some sophisticated definitions (length of arc, which needs a definition of local distance that must then be integrated). That (directed) area can be defined in a vector space and arc length cannot supports this idea, though this is not sufficient, since the definition of radius becomes a casualty. Anyway, this makes the the idea of using area sound more natural. I expect that both the arc length and area approaches work in the Euclidean and hyperbolic angle cases (arc length, in the hyperbolic case, is clearly not the Euclidean length).

However, if we are to discuss approaches with this depth, it would be nice to find a more general definition of angle. For example, the angle of intersection of two (sufficiently smooth) curves in any conformal space is well-defined, but none of the usual definitions of angle would appear to apply here because there is no concept of length or area that can be defined. —Quondum 22:23, 12 August 2025 (UTC)[reply]

Well I've come around to the view that maybe this article can be broader than the Euclidean definition and we can discuss other concepts. Essentially the first half of the lead and the first few sections should discuss the Euclidean definition and be written to a middle school level and then the rest can be as technical as necessary.

Anyways with regards to the generalizations I was thinking that we might have a section generalizations of the angle sort of like the article generalizations of the derivative Mathnerd314159 (talk) 04:38, 13 August 2025 (UTC)[reply]

Angle defined

A copy of the algebra and calculus, that Hardy used to define angle as an area expressed by an integral, is now found in the article on A Course of Pure Mathematics. — Rgdboer (talk) 01:09, 16 August 2025 (UTC)[reply]

Definition of angle

The definition of angle has been discussed above, but I don't think we've quite got the right solution yet or reached any consensus. The previous definition talked about the "opening" between two lines being the angle, which is one of a number of valid perspectives, but arguably not the most common view.

I propose the lead to start with: "In geometry, an angle is formed by two lines in the same plane that meet at a point".

This is a general description rather than a precise definition, but I think it's a fair starting point for the reader and consistent with most common specific definitions. The article would then benefit with at least a short section providing and contrasting some of the specific definitions.

Saxon (Algebra I: An Incremental Development, page 282) follows this approach:

European authors generally define an angle to be the opening between the rays. Thus, to them the angle is the set of points bounded by the rays. American authors tend to define the angle to be the rays themselves. To them the angle is the set of points that make up the rays. Others say that the rays are the sides of the angle but don't say what the angle is. Some don't speak of the opening at all, but define an angle to be a rotation of a ray about its endpoint. A precise definition is not required in this book so we will just say

An angle is formed by two half lines or rays that are in the same plane and that have a common endpoint.

TheGrifter80 (talk) 15:00, 8 September 2025 (UTC)[reply]

I think it's essential that we immediately lead with both of the main classes of definitions of angle: (1) a geometric figure formed by two rays, or (2) a quantity expressing the separation between two rays or the extent of a rotation. These two concepts are distinct but closely related, and each has several variants which appear in various contexts. –jacobolus (t) 15:46, 8 September 2025 (UTC)[reply]

(There is also (3) "angle measure", a real number representing the object from definition (2)). One of the early sections of this article can then un-pack some of the most common variants and describe their relationships, ideally with lots of pictures. A later section toward the bottom can describe some of the less common variants and generalizations to e.g. curves in the plane, geodesics on curved surfaces, intersections between higher-dimensional flats, solid angle, and so on. –jacobolus (t) 16:21, 8 September 2025 (UTC)[reply]

I agree with all you have said...feel free to edit what I've done to reflect that. TheGrifter80 (talk) 16:43, 8 September 2025 (UTC)[reply]

I think we should probably also bold "plane angle" in the first sentence (rather than just "angle"), and mention at least dihedral angle and solid angle further down the lead section.

Aside: Euclid has: η΄ Ἐπίπεδος δὲ γωνία ἐστὶν ἡ ἐν ἐπιπέδῳ δύο γραμμῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐπ᾿ εὐθείας κειμένων πρὸς ἀλλήλας τῶν γραμμῶν κλίσις. [My paraphrase from various translations: 8. A plane angle is the inclination to one another of two [straight] lines in a plane which meet but do not coincide; note that for Euclid "straight line" here means what we call a "line segment"] –jacobolus (t) 17:00, 8 September 2025 (UTC)[reply]

I can assure you that κλίσις is better translated as "opening" than "inclination". But it's academic, since you removed it, along with the clarification that this was a Euclidean angle. I'm still bitter about it. It seems from the Saxon source that the "opening" definition is in fact the most common. I don't think Saxon is right though, the opening between the rays is not equivalent to the set of points bounded by the rays, because that would be the angle's sector. Cue the Blind men and an elephant parable. Mathnerd314159 (talk) 05:33, 9 September 2025 (UTC)[reply]

Who is "you" here? To be clear: you are "bitter" that I think we should have an article whose scope is the general geometric concept of "angle", and I thus changed "in Euclidean geometry" to "in geometry" as the first few words? Various concepts under the name "angle" are applied much more broadly than Euclidean geometry per se, and to me the narrow scope seems very unhelpful for developing a complete article about the topic. –jacobolus (t) 06:11, 9 September 2025 (UTC)[reply]

I don't speak Ancient Greek, but if I look this up in wiktionary, wikt:κλίσις, inclination is the first definition given, apparently descended from a proto-indo-European word for to lean, slope, or incline, also related to various words meaning "to bend". Fitzpatrick here has: κλίσις -εως, : no, inclination, bending. Heath renders Euclid's definition as "A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line." –jacobolus (t) 06:13, 9 September 2025 (UTC)[reply]

Heath actually has quite a lot to say about the definition of angle, and the topic of various ancient definitions would be worth discussing in some detail in this article. I guess Euclid's definition #8 was supposed to apply to intersections between curves, so I shouldn't have added [straight] above; I was confused about the part about not lying on a straight line, but perhaps should not assume that means only "do not coincide". (I am by no means an expert in the history or languages here, and it's not precisely clear to me what all of these various definitions, as translated into English, are intended to mean.) –jacobolus (t) 06:30, 9 September 2025 (UTC)[reply]

As for the point, straight line, and plane, so for the angle, Schotten [1890] gives a valuable summary, classification and criticism of the different modern views up to date (Juhalt und Methode des planimetrischen Unterrichts, 1. 18.3, Pp. 94—183); and for later developments represented by Veronese reference may be made to the second article (by Amaldi) in Questioni riguardanti la geometria elemantare (Bologna, 1900) already referred to.
With one or two exceptions, says Schotten, the definitions of an angle may be classed in three groups representing generally the following views:
The angle is the difference of direction between two straight lines. (With this group may be compared Euclid’s definition of an angle as an inclination.)
The angle is the quantity or amount (or the measure) of the rotation necessary to bring one of its sides from its own position to that of the other side without its moving out of the plane containing both.
The angle ts the portion of a plane included between two straight lines in the plane which meet in a point (or two rays issuing from the point).

It is remarkable however that nearly all of the text-books which give definitions different from those in group 2 add to them something pointing to a connexion between an angle and rotation: a striking indication that the essential nature of an angle is closely connected with rotation, and that a good definition must take account of that connexion.

(p. 179) –jacobolus (t) 06:49, 9 September 2025 (UTC)[reply]

To understand your perspective, when you say this article should be about Euclidian angles, do you mean angles as characterised by Euclid himself or do you mean angles in what we call Euclidian geometry (as opposed to non-Euclidian geometry)? TheGrifter80 (talk) 01:32, 10 September 2025 (UTC)[reply]

Sorry this reply was meant for Mathnerd's comment. TheGrifter80 (talk) 01:34, 10 September 2025 (UTC)[reply]

Yes, I'm bitter. A good Wikipedia article defines its scope in the lead. In cases where a term has multiple meanings, there are multiple articles. For example with "Joker" we have Joker (Jack Napier), Joker (The Dark Knight), Joker (Mass Effect), Joker (comic strip), etc. There is quite enough material just for an article on the concept of Euclidean angle, and your campaign to add alternative definitions of the quantities of angles and non-Euclidean angles and so on is in my view an attempt to mix drinking water with dirt. It is not dangerous, but it produces a muddy mess, and you certainly can't drink it anymore. Less prosaically, the Euclidean definition has endured for thousands of years, so any other definition is likely flawed. The narrow scope is exactly what is needed to write a compelling article; an article about everything is an article about nothing in particular. And in practice, what is necessary in grade schools is an article for the Euclidean definition.

And yes, I consulted those sources. There is also Proclus and this paper (which discusses the ancient Greek somewhat) and EB1911 and Frankland and this paper on horn angles. And of course wiktionary. I'd be OK with "bend" too, but opening is more commonly used in phrasing.

I also looked at a bunch of other translations, I don't remember which books they were exactly, but the last one may shed some light on your confusion about colinearity:

A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.
A plane rectilineal angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same "direction".
A plane angle is the inclination of two lines to one another, in a plane, which meet together, but are not in the same direction.
A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
A plane rectilineal angle is the inclination of two straight lines to one another in the same plane, which lines meet together, but do not lie in continuation of each other.

Mathnerd314159 (talk) 23:48, 9 September 2025 (UTC)[reply]

Euclid directly says they don't lie in the same straight line; the reason this is confusing is because he splits the definition into 2 parts, definition #8 about "plane angle" between lines (i.e. curve segments), and #9 about "rectilineal" plane angles between straight lines (i.e. line segments). I think Heath may have the right idea that the definition was really for the straight case but then was split into two parts without really explaining the idea clearly or maybe without sufficiently thinking it through. It's not clear whether this caveat is supposed to apply to e.g. two curve segments which coincide or if that's supposed to be excluded; if the latter, it's not clear whether e.g. the "angle" between a curve and its tangent line is supposed to be considered an "angle" or not. Later authors also found this definition confusing and tried to come up with alternative definitions which would be clearer or more explicit.

I don't think it's useful for us to have separate articles called angle, angle (between curves in the Euclidean plane), angle (space curves), angle (abstract contexts), angle (affine geometry), angle (Lorentzian plane), angle (between generalized circles in conformal geometry), angle (hyperbolic plane), angle (spherical), angle (on a general surface), rotation angle, etc. Even if we come up with enough information about one of these variant or generalized contexts to deserve a separate article (which could plausibly happen for one or more of these), we should at least mention it somewhere in the main article titled angle, in just the same way that our article distance has a comprehensive scope, and covers most of the main specific types of distances at least in summary.

Perhaps more to the point here, I don't think it's useful even when discussing Euclidean angles per se for us to have separate articles about angle (geometric figure), angle (inclination between lines), angle (subset of the plane) and angle measure; these several concepts are definitionally distinct but are intimately inter-related, commonly conflated even by professional mathematicians and sometimes within the same sentence, and cannot be explained independently without extensive discussion of each-other. –jacobolus (t) 23:56, 9 September 2025 (UTC)[reply]

EB1911 seems like a reasonable precendent. It has "ANGLE [...] in geometry, the inclination of one line or plane to another." It explicitly discusses multiple concepts of plane angle, as well as solid angle, dihedral and polyhedral angles, spherical angles, curvilinear angles, and some others. –jacobolus (t) 01:10, 10 September 2025 (UTC)[reply]

Some further comments on this point. I think the distinction between (1) and (2) you make is useful and clearly articulated. A couple of observations.

When angle is defined as a figure (class 1), the figure still has a quantity/measure (class 2), and the word "angle" is sometimes used for both of these things in that context. ie it's common to see∠ABC = 90%, and this is understood even thought it would be more accurate to say m(∠ABC) = 90%. So even when explicitly defined as a figure there is still some ambiguous use of terminology, and angle can be both (1) and (2).

But for other definitions of angle (inclination, separation, rotation, area of the plane) there seems only to be the quantity/measure aspect (class 2).

So then a question: is the important point to make in the lead to (a) highlight the imprecise/ambiguous use of the word "angle" to mean two different things even given a specific concept of angle or (b) distinguish between different underlying definitions / concepts of what the angle actually is?

Or is this just splitting hairs and (a) and (b) are actually more or less the same thing? TheGrifter80 (talk) 13:16, 9 September 2025 (UTC)[reply]

I think these various concepts are substantially different and it's worth mentioning in the lead and unpacking more extensively later in the article, with at least a few paragraphs about it somewhat near the top. –jacobolus (t) 14:58, 9 September 2025 (UTC)[reply]