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One of the 500 most frequently viewed mathematics articles.

Cone (solid) vs Right circlar cone[edit]

Now there seems to be an issue with cone (geometry) and right circular cone. I'm inclined to redirect the latter here; most of the links in require the 'right circular' meaning of cone.

Charles Matthews 18:54, 5 Jul 2004 (UTC)

Where was "here"? The talk for cone (geometry) redirects here.
--Jerzy (t) 20:09, 2005 May 4 (UTC)

Cone (geometry) vs Cone (linear algebra) vs Cone (solid)[edit]

This section is obsolete now --Jorge Stolfi 03:51, 23 December 2005 (UTC) The redirect links between these articles don't really make sense. The more abstract definition of cone is the following: A subset of a vector space is a cone if implies for any .

The article for Cone (solid) seem to be for a right cylindrical cone in 3-space. Since this is a particular instance of a cone, so it probably should be listed as an example under Cone (geometry).

Also, I think the definition for pointed cone is incorrect. Counterexample: R^3 is a cone that contains the origin. It is not pointed.

I don't know the definition of pointed for a general cone. For a polyhedral cone, I believe pointed means that contains an extreme point. --Scott 17:45, 12 August 2005 (UTC)

In the literature (e.g. Schrijver's "Theory of Linear and Integer Programming"), a Polyhedron is called pointed if its lineality space has dimension zero, i.e. = {0}. -- 14:41, 8 June 2011

Cone (geometry) vs Cone (solid)[edit]

WARNING - this section does not make sense any more because some well-meaning soul "fixed" the original links so they are all "cone (geometry)" now. Anyway the move happened and this section is moot. --Jorge Stolfi (talk) 09:38, 8 February 2010 (UTC)

Until 2005/dec/22, the article cone (geometry) was about a concept of linear algebra, and the common geometrical shape was described in cone (geometry). There were may pages pointing at the former when they should point to the latter. So I renamed cone (geometry) to cone (linear algebra) and changed all links to cone (geometry) to point to the appropriate article. Now cone (geometry) points back to the cone disamb page.

But now there still one inconsistency: articles on geometric solids are usually named xxx (geometry), not xxx (solid). Perhaps we should now delete cone (geometry), move cone (geometry) to cone (geometry), and fix again all the links... Jorge Stolfi 03:51, 23 December 2005 (UTC)


Add *Support or *Oppose followed by an optional one-sentence explanation, then sign your opinion with ~~~~
  • Support --Swift 07:39, 30 June 2006 (UTC)
  • Support --Usgnus 13:51, 28 July 2006 (UTC)
  • Support. – Axman () 09:44, 30 July 2006 (UTC)
  • Support --Edgelord 06:13, 31 July 2006 (UTC)


I see why my links to and were deleted on this page. However, by Wikipedia's own definition of spam, my links are clearly not spam, as there is no advertising or commercial products being sold.

I taught math for 13 years. There's substantial educational practice that students need access to auxiliary information beyond the classroom to understand a topic. Also, my site offers free dynamic quizzes to help students learn.

I do understand that there are rules for linking to one's own website. If someone finds this section of my site relevant, please link to it.

Why is the volume 1/3 of the volume of a cylinder?[edit]

i am just wondering how did anyone justify as it is? we cannot really assume anything. is there really any proof to it? in a triangular pyramid we can still prove it. how could we prove it on a cone?

This result goes back to Archimedes. If you believe it for a pyramid, just put the cone inside a cylinder, then slice the cylinder into a gazillion wedges like a panettone. (But anyway in WP one does not give proofs, just results.) Jorge Stolfi 09:54, 21 February 2006 (UTC)
See as this bears out experimentally.
Im knew to wikipedia editing (long time reader) so please delete this if its out of line etc... 1/3 can be proven using calculus by integrating a function r(z)(radius wrt height) over the interval (0, h),this definite interval yields the formula for the volume of a cone. —Preceding unsigned comment added by (talkcontribs) 19:20, 25 August 2006
No, I wouldn't say out of line at all. Such discussion, however, usually takes place on the Reference desk. The talk pages being for discussing article content. Glad you've deceided to test the edit button :-). --Swift 20:42, 25 August 2006 (UTC)

Page moved[edit]

This article has been renamed as the result of a move request. Vegaswikian 17:18, 4 August 2006 (UTC)

Merge proposal[edit]

I've put up a proposal to merge Right circular cone into this article. The RCC is a special case of a cone and I don't think it needs a page for itself. --Swift 07:32, 30 June 2006 (UTC)

I've added Conic solid, Projective cone and Conical surface to this proposal. --Swift 20:19, 17 August 2006 (UTC)
Right circular cone should definitely be merged, but I'm not sure about the others. The combined article would need some major restructurting, but that might be a good thing. Let's try it. —Keenan Pepper 01:59, 18 August 2006 (UTC)
Strong support, these pages are a mess. I might do this myself if I get time. -Ravedave (help name my baby) 17:39, 26 September 2006 (UTC)
I've done a sloppy merger and added a {{cleanup}} tag. I didn't merge the Projective cone since that article is quite a bit more mathmatical than I think a *_(geometry) article should be (I wasn't even sure if the sub-section on the parametrization of the conic solid was too much). If everyone is happy, I'll AfD the merged ones in the next few days. --Swift 05:51, 28 September 2006 (UTC)
Don't AFD them, they should be redirected per Wikipedia:Merging_and_moving_pages. I have redirected Conic solid. I removed the template from Projective cone and improved that article a bit. It looks like Conical surface still needs to be worked in and then redirected. -Ravedave (help name my baby) 01:24, 29 September 2006 (UTC)

I undid the merge of 'conical surface' into 'cone (geometry)'. The two concepts are as different as spheres and circles. Most concepts (axis, base, radius, directrix, frustum, volume, etc.) either are defined for only one case, or require a separate definition for each case. Merging the two articles saves very little text and creates a lot of confusion. Jorge Stolfi 15:22, 30 September 2007 (UTC)

Surface area of a cone[edit]

In my math textbook, it says that the surface area of a cone is found using this formula: A=(pi)rl, where l is the slant height, r is the radius of the circular base, and A is the surface area of the cone. I am not sure if this is just for right circular cones, but the area formulas in the article seem a little complicated... this is what the article says: , where is the slant height. and S(t,u) = (ucosθcost,ucosθsint,usinθ), if any of you guys know a simpler way of expressing this which most people will understand, please do... 19:23, 23 March 2007 (UTC)

The formula is the area of the side. is added to give the total surface area, which includes the area of the base. Also, s here means the same thing as l in your textbook. Finally, I'm putting your math formulas in the math style, and I hope it's okay with you. Generalcp702 18:31, 31 March 2007 (UTC)

I think it is true that from any isosceles triangle possible to made cone. So then like is base b and l or s is altitude h, so surface area is .

types of cone[edit]

"There are four types of cones: circular, elliptical, right, and oblique, all of which are conic solids."

These seem not to be mutually exclusive types. e.g. a "right circular" is a valid cone type. Is there some way this could be better explained.

"All pyramids are also cones"

Which type are they? or are they a 5th type? In my head saying that a cone is just a n = ∞ cone makes sense, but I am aware that standard terminolgy is not always logical. The pyramid page just says

"[a pyramid]is a conic solid with polygonal base"

So is conic solid the parent class to which cone and pyramid belong? -- 09:56, 8 May 2007 (UTC)

"In general, a cone is a pyramid with a circular cross section." -- 10:08, 8 May 2007 (UTC)

We should state the generalization but also include a pyramid as a conic Nicholas SL Smith 17:35, 7 June 2007 (UTC)

All cones are pyramids. Not all pyramids are cones. Specifically, cones are pyramids with circular bases, i.e. a special case. This is sourced here [1]. If necessary, I have a source that indicates this source is not just reliable but published by a highly respected source in the mathematical community.

  • Last month I found an error in that "highly respected source", so there. 8-)
    Anyway, there is hardly ever *the* definition of a math concept; there are usually many definitions. Even such a basic thing like natural number has two definitions, both still in use.
    Normally each author (including mathworld's) picks one definition and pretends the others do not exist. But Wikipedia cannot do that; it must record all definitions and not try to make value judgements.
    The use of "cone" for shapes with non-circular section is actually common. Just google "non circular cone" or "elliptical cone" to get an idea. In particular I liked this example which shows that non-circular cones are not only popular but even legal! 8-)
    All the best, --Jorge Stolfi (talk) 09:56, 8 February 2010 (UTC)
    • I have no objection to calling a cone a fruitcake, so long as it comes from a reliable source. Since this is a subject in the academic field of geometry, all assertions should be come from reliable, academic sources. I've provided an academic source that defines a cone. If you wish to contradict or expand this, please use another reliable source. A patent registration firm is not equal to an academic source when it comes to defining mathematical terms. Ultimately, this article needs much better sourcing, as it had none at all until today. Rklawton (talk) 13:48, 8 February 2010 (UTC)

Speaking of which "google it and you will see" is not a substitute for reliable sources. Rklawton (talk) 17:27, 8 February 2010 (UTC)

  • Hi Rklawton, first please note that this article is about a concept of "(geometry)", not "(academic geometry)". Geometry is widely used outside academia — by engineers, builders, technicians, and even by lawyers it seems. Also, Google is a good way to *find* sources; in this case we are tryng to establish "what do people understand by 'cone' as a geometric shape", so the search should not be confined to academic sources. That, by the way, is what dictionary makers (the OED specifically) do when they want to know what words mean.
As for that, Here are some sources that apparently assume "elliptic cone" is a known concept:
And finally (I wouldn't have guessed, Google found it for me)
So, well, I still defend my sentence. All the best, --Jorge Stolfi (talk) 23:49, 8 February 2010 (UTC)

I'm not against elliptical cones. I'm in favor of using sources throughout the article so that people using it can be reassured the article wasn't authored by an eight-grader. I also prefer viewing a cone as special case pyramid and not the other way around, but would be pleased to present a reliably sourced, differing view. Rklawton (talk) 03:09, 9 February 2010 (UTC)

Other Mathematical Meanings[edit]

I'm no math wiz, but it seems to me that the definitions given using lines and half-lines passing through a common point are lacking some fundemental constraint. It is possible to have an infinite variety of lines passing through a single point that have nothing to do with the cone - for example, the cone's axis will also pass through this point and yet is not part of the set that defines the cone.

Pleah (talk) 17:37, 9 February 2010 (UTC)

the axis if any[edit]

The axis of a cone is the straight line (if any), passing through the apex, about which the base has a rotational symmetry.

If this means that only a right circular cone has an axis, better to say so explicitly. —Tamfang (talk) 01:17, 25 March 2012 (UTC)

No, any symmetric cone. A general cone doesn't have to be circular but it can still have rotational symmetry and so an axis. In common usage 'cone' means right circular, as described in the next paragraph. The article is a bit untidy, reading it: it doesn't make the distinction between a right circular cone and a more general one clear later on.--JohnBlackburnewordsdeeds 02:10, 25 March 2012 (UTC)

Move discussion in progress[edit]

There is a move discussion in progress which affects this page. Please participate at Talk:Cone - Requested move and not in this talk page section. Thank you. 22:49, 24 July 2012 (UTC)

Move discussion in progress[edit]

There is a move discussion in progress on Talk:Cone which affects this page. Please participate on that page and not in this talk page section. Thank you. —RMCD bot 18:45, 31 July 2012 (UTC)

Request for visual clarification[edit]

I'd like to formally request anyone with the skills to show the derivation of the volume and surface area by means of visual diagrams.

It's very easy for people with great maths skills to relate to formulae, but without a clear understanding of just how the various arguments relate, we all must take this page on face value: that's not a particularly helpful method of helping people understand concepts. OK, I can assume that the symbols are correctly expressed and arranged, but it still doesn't help me understand the relationship between (say) a right pyramid and the cone it's being compared with to derive the equivalence. This is especially true when all the backreferences followed (in my case, sphere-cone-Cavalieri's method-method of exhaustion) have no (or very few) visual cues to permit those of us with poor mathematical skills to understand the relationships. There's no "aha!" moment, in other words.

I understand and I absolutely appreciate that WP isn't a teaching resource per se; but it would help many people to visualise the concepts, and that cements the ideas far better than rote cutting and pasting (or rehearsal) of "dry" formulae.

I suspect there would be more benefits than drawbacks. And I can say, if I could, I absolutely would, but I still can't (even at 49 years of age!). So anything that would help an idiot like me would help other people too, I'm sure. Comments? Cephas Borg (talk) 06:21, 26 October 2012 (UTC)


The picture concerning the cone volume seems rather dubious i suggest to change it to a more fitting one. There is no need to post pictures of people on a wikipedia page concerning a mathematical problem, this is not facebook.— Preceding unsigned comment added by (talk) 15:44, 27 May 2013 (UTC)

The photo with the volume formula is helpful -- most "readers" do not read the text but merely look at the photos and diagrams so it is helpful in this regard. The volume formula is correct and instructive. The photo-diagram helps the article be more visually appealing, and does not promote any person or business or cause and no reader will confuse this article with Facebook.--Tomwsulcer (talk) 16:10, 27 May 2013 (UTC)

Though I agree with your premise, I feel like the referenced picture could be switched for one more fitting to the article. Personally, I came to this page and found myself confused because I didn't look at the picture and the formula posted lists B as the area of the base while many engineers and the like consider B to be shorthand for the base length. So I actually feel that, although the volume formula is correct, it would be better to list the equations along the lines of:

Then A could be defined as the area of the base for clarity (πr2). — Preceding unsigned comment added by (talk) 19:12, 16 July 2013 (UTC)

I think the picture is very unfortunate because almost all of its space is taken up by an irrelevant person, and the relevant part -- showing the area formula for the cone and the location of the formula's components in an actual visual cone -- is relegated to a small part of the image. But the relevant part is very appropriate for the article. I hope someone can reproduce the relevant part in a new image and use that one to replace the existing image. Duoduoduo (talk) 15:19, 11 September 2013 (UTC)
I guess it's supposed to be whimsical, but it fails. It is terribly distracting from an article about geometry.-- (talk) 21:23, 13 December 2013 (UTC)

Does a cone have to have a circular cross-section?[edit]

Let's assume yes for a moment.

Then why is it not made clearer in the lead section that the base must be circular or elliptical?

Why does it say in the lead section that the axis is the straight line "if any" ... etc.? Doesn't every cone with a circular cross-section have an axis?

The explanation of "right circular" seems to have been written under the assumption that these properties can be independently applied, which is not possible with a circular cross-section, when one forces the other.

I think it is all a bit muddled, but the first thing is to verify that cones definitely must have circular cross-sections. (talk) 02:44, 21 December 2013 (UTC)


The article ought to define exactly what is meant by "base" in the context of a cone. Is the base the directrix itself ("circular means that the base is a circle") or the plane surface bounded by the directrix (as suggested by the formula ) or both? Is base in this context the same as base (geometry) meaning the face of a solid? Is a cone therefore per definition solid? Isheden (talk) 15:28, 23 January 2014 (UTC)

Merger proposal[edit]

I propose Slant height be merged into this page. The article doesn't include information beyond definitions provided here. (talk) 03:26, 11 June 2014 (UTC)

Request of putting in simple Proof of enclosed volume using disc method[edit]

My first post in wiki, i just wanted to know if my proof would be applicable here. First we assume a cone to be built out of several Zylinders, knowing the volume of that already

V= pi*r²*h So we now divide the cone into cylinders with linear declining radius. Dividing the cone into an n amount of cylinders with each of equal hight this leaves us with the sum of Volumes of cylinders.

So for example:

V=1/5 pi/(5/5*r)²+1/5 pi*(r/5*4)²+1/5 pi*(r/5*3)²+1/5*pi*(r/5*2)²+1/5*pi*(r/5*1)²

taking out constant factors we are left with

V=1/5*pi*5²*r²*(1²+2²+3²+4²+5²) V=1/5³*pi*r²*(1²+2²+3²+4²+5²)

so in general this is V=1/n³*pi*r²*(k=1 to n ∑ k²)so sum of k² which can be substitued by (see 1/3 n³ + 1/2 n² + 1/6 n so we are left with V = pi*1/n³*r²*(1/3n³+1/2 n²+1/6n) which multiplied out is V= 1/3*pi*r²+1/2n*pi*r²+1/6n²*pi*r² so now it's esay to see that as n approaches infinity (so the number of zylinders we built our cone out of) the volume approaches 1/3*pi*r² which is as we know the volume of our cone

So math wiki community, could we implement this proof into this article? I think it is esay to understand and has a very awsome connection to the summation of square numbers, and would strongly appeal to the public.

With kind regards, Daniel — Preceding unsigned comment added by Couldbeanyonereally (talkcontribs) 13:50, 1 May 2015 (UTC)

algorithm to find apex of a cone?[edit]

Say I have a function f(x,y) such that, for any x,y, it returns a point on a cone. What is the best numerical algo for finding the apex of this code? This seems to be trickier than one might first imagine, due to numerical instabilities. (talk) 20:56, 18 April 2016 (UTC)

This is not a forum for discussions of all things related to cones (even genuinely interesting questions like yours), it is a page to discuss edits to the associated article. You might try the WP:Reference desk instead. --JBL (talk) 22:41, 18 April 2016 (UTC)

Polygonal cone?[edit]

Currently the lead says that the base of a cone is frequently, though not necessarily, circular. It then goes on to say

A cone with a polygonal base is called a pyramid.[2]

But reference [2] does not say that. It says

Conic solids have but one base. Pyramids have lateral edges which connect vertices of the base polygon with the vertex. In a cone, the lateral edge is any segment whose endpoints are the vertex and a point on the base circle. The triangular, non-base, faces of a pyramid are lateral faces. Pyramids and cones can also be....

clearly distinguishing between pyramids and cones.

Is there any source for the assertion that something with a polygonal base and an apex can be called a cone? I doubt it, and I think the assertion should be either sourced or deleted. If deleted, I think we should find a source that the base of a cone has to be a circle or ellipse, and mention that in the article. Loraof (talk) 21:11, 3 May 2016 (UTC)

Are you expressing doubt that people use the word "cone" to refer to certain polyhedra? Grünbaum's Convex Polytopes defines on p23 of the second edition (paraphrasing notation lightly)

a convex set C is a cone with apex 0 provided ax is in C whenever x is in c and a ≥ 0.

This certainly contains the polygonal case (in fact it is the case of most interest in the book).
On the other hand, I would not propose using this definition early in this article. --JBL (talk) 21:23, 3 May 2016 (UTC)
Okay, I'll put that reference in there. Could you take a look at the article Conical surface and see if it needs some corrections? It says
In three coordinates, x, y and z, the general equation for a cone with apex at origin is a homogeneous equation of degree 2 given by
which looks to me like it puts a restriction on the base which excludes polygons and other non-quadratic shapes. Thanks. Loraof (talk) 00:05, 4 May 2016 (UTC)
A few observations: first, I agree, that equation is certainly about the surfaces that form the boundaries of only a proper subset of the things called "cones" in this article. (Though exactly which subset is not obvious to me.) Second, the phrase "conical surface" is very much about 3 dimensions, and Grünbaum is certainly contemplating cones in arbitrary dimension (and this agrees with the usage that is common among combinatorialists who study polyhedra, in my experience), so the phrase "conical surface" wouldn't come up in that context (you'd expect "boundary" or something instead). Third, this article is really hard to write well, because these more general definitions of cone (different bases, possibly unbounded, possibly in other dimensions) are not the common one (circular, bounded, three-dimensional). I do not really have a conclusion here; I will try to look at conical surface tomorrow or the next day. --JBL (talk) 01:56, 4 May 2016 (UTC)
I think we should stick to textbook definitions. I feel uneasy about some of the wording that is now in this article and in Conic section. For example: "if the base is right circular" How can a base be right circular? "an infinite or doubly infinite cone" Aren't all cones, by definition, infinite in both directions? In the article about Conic section: "the surface of a double cone" Isn’t a cone, by definition, a surface? "the intersection of the boundary of a cone with a plane" How does a cone have a boundary? — Anita5192 (talk) 04:35, 4 May 2016 (UTC)
While I agree with Anita5192's comment about sticking to textbook definitions, this can be a bit trickier than we would like. I have come across texts that distinguish pyramids and cones (one has "flat" lateral surfaces and the other doesn't); ones that define a directrix to be a curve (so, in general, these cones are two dimensional) and others that permit it to be a region (giving solid cones); elementary treatments that seem to think that all cones are right circular cones and others that grudgingly permit more general quadratic cones (BTW that equation in conical surface is a general equation for a quadratic cone whose directrix is a conic section ... but only over fields whose characteristic is not two, so not all that general from my point of view). I would say that as soon as you loosen up what you allow a directrix to be, you'll have to call pyramids cones. Deciding which textbooks to follow will be difficult (and presenting all points of view would be massively confusing). My personal favorite definition of a cone is the union of lines joining the apex with a point on the directrix (with a discussion of possibilities for the directrix). This is even more general than Grünbaum's definition since he only takes rays instead of full lines. Bill Cherowitzo (talk) 05:32, 4 May 2016 (UTC)
@Anita5192: you say twice "by definition", but by Grünbaum's definition neither of these things is true. (Although I also note that the lead section of this article explicitly restricts to dimension 3, so in some sense the polytopal/convex geometry definition is about a different kind of cone.)
Tentatively, it seems to me like the article should have a high-level structure with at least the following subdivisions: three-dimensional circular cones; three-dimensional cones with arbitrary bases; and higher-dimensional cones. --JBL (talk) 14:42, 4 May 2016 (UTC)

Given that we need to be neutral in the face of a variety of definitions in the literature, I propose this as a draft of a new lead:

In three-dimensional space in mathematics, a cone is formed by a set of line segments, half-lines, or lines connecting a common point called the apex to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a circle, any one-dimensional quadratic form in the plane, any closed one-dimensional figure, or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a solid object; otherwise it is a two-dimensional object in three-dimensional space.
In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends infinitely far. In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone.
Cones can also be generalized to higher dimensions.

Loraof (talk) 16:07, 4 May 2016 (UTC)

Generating line[edit]

In Conic section article there is mentioned "Generating line". In cone article it is called "generatrix". Would it be correct adding (mentioning) "Generating line" as synonym? generating_line_anon_user 11:26, 17 Nov 2016 (UTC) — Preceding unsigned comment added by (talk)

I like this suggestion and have implemented it. --JBL (talk) 14:41, 17 November 2016 (UTC)