# Talk:Sphere

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## Bug in new wiki software

I'm thinking there is some bug with the new wiki software, I saw the text went away that Zundark is talking about, but couldn't figure out how to bring it back. I'm using internet explorer 5, and had no problems with the old software. Either that, or my dog ate it. <grin> -- BenBaker

There was such a bug on the first day. Should be long gone. For old version, you'll have to wait for the promised http://old.wikipedia.com --Magnus Manske
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## Sphericon

Does anyone have any objections to the following text being added...?

A sphere can also be defined as a sphericon that is based on a polygon approaching a circle.

Proberts2003 19:31, 10 Apr 2004 (UTC)

I do Tosha 00:42, 11 Apr 2004 (UTC)

Which are...? Proberts2003 01:45, 12 Apr 2004 (UTC)

Ok, I think it is not directly relevent, one can mention sphere in sphericon, but not other way arround. (Otherwise you should include in this article ref to all geometric topics) Yet an other thing: yes it can be defined this way but it would be most wierd way to define sphere. Tosha 05:15, 12 Apr 2004 (UTC)

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## Water drops

The article currently says "water drops (in the absence of gravity) are spheres". I changed this to simply "small water drops are spheres" because photographs of rain show small spheres (see http://www.ems.psu.edu/~fraser/Bad/BadRain.html ). -- DavidCary 23:33, 27 Jun 2004 (UTC)

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## Conventions

I believe the claim that the topological convention is "the most common" is NPOV, as I believe it's a true statement. (Of course, people who work in the fields using the other definition will feel differently...I don't directly work in either field, so I'm going by my own experience. The topological definition is the only definition I've ever seen or heard; until today I wasn't aware there was another convention.) It would be good if the actual fields of research that use this convention were spelled out. (I honestly have no idea what these are.) I strongly object to the use of the term "geometrical definition" or "geometrical convention", because "geometry" is too broad a word; also, it conflicts with the convention used by differential geometers, who most certainly consider themselves to study "geometry". Revolver 00:08, 12 Jul 2004 (UTC)

Unless someone can provide at least one example of a peer-reviewed mathematical paper that uses the term "n-sphere" to mean (n-1)-sphere, then I'm going to remove all this stuff about different conventions. There is only one convention, as far as I'm aware. --Zundark 09:35, 23 Dec 2004 (UTC)
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## Sections

Can we split this up into two sections — Geometry and Topology — as I did for Ball (mathematics)? In the Geometry section have the metric-space definition (locus of points a radius from the center; boundary of a ball), the Eulcidean examples, the current Equations subsection, and all that jazz; and under Topology have

A sphere is any space homeomorphic to the Euclidean sphere described above under Geometry.

, definition of n-sphere, the fact that the boundary of a ball is a sphere one dimension down (for n>0), and definitions of homology sphere.msh210 21:17, 27 Oct 2004 (UTC)

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## Proof of compactness

the prove that a n-sphere is compact is not complete?

• needs why complement has only innerpoints
• why is it bounded?? (maybe due it's definition $S^n = \{ x \epsilon R^{n+1} |d(x,0) = 1\}$)
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## Split article?

Currently the material on n-spheres is split between this article and hypersphere. The situation is somewhat unsatisfactory. I propose separating the material along logical lines

• the sphere article should focus the ordinary 2-sphere in Euclidean 3-space. This is probably what people expect when they type sphere into the search box.
• the n-sphere article should discuss the general case, with sections on both geometry and topology. We can have a redirect from hypersphere to there (I prefer it this way since the term hypersphere is not in common usage in mathematics).

I'm happy to do this split if no one objects. Comments? -- Fropuff 02:29, 2005 Apr 16 (UTC)

Iff sphere has a link to n-sphere at its top (not just {{otheruses}}), I agree. That is, it should say something like For higher-dimensional spheres in mathematics, see n-sphere; for other spheres see Sphere (disambiguation).msh210 13:50, 17 Apr 2005 (UTC)
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## Perfect symmetry

Seems to me that the sphere is not perfectly symmetrical. For example, punctured 3-space is also self-similar under radial contractions, and so is more symmetrical. Ideas of perfection led to a lot of wrong-headed ideas about e.g. planetary motion. Maybe they are worth mentioning, but in my opinion putting "perfection" up front doesn't help. --JahJah 02:50, 22 August 2005 (UTC)

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## Convoluted

Definition number one from http://www.tfd.com/sphere It's succinct, accurate, and readable. It accomplishes in one sentence what Wikipedia's entire article fails. To whom are you trying to explain a sphere? Who is your audience? If your son or daughter asked you what a sphere is, how would you describe it?

Also, ease up on the passive voice.192.165.166.4

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## finding the surface area of part of a sphere

I think it would be very useful to have either a derivation for, or simply the integral one needs to use to find the surface area of part of a sphere. Also perhaps it would also be useful to do the same thing for the volume of part of a sphere. 68.6.112.70 18:22, 10 May 2006 (UTC)

I would like to echo and amplify this comment. After about an hour of google searching I didn't find an article I liked deriving the surface area of the sphere. It seems this and the volume derivation should be front and center. Particularly, an elementary derivation would be nice. Kaimiddleton (talk) 21:57, 25 December 2008 (UTC)
I empathise with you. As a pupil, it was never explained to me why the surface area of a sphere is what it is, although the volume formula was, as was the area of a circle. I offer a relatively simple explanation to those not so comfortable with double integrals and spherical coordinates. Unfortunately I cannot cite any references, as I don’t have any (I’ve never actually seen an explanation in any book). Perhaps someone else can verify and/or provide some diagrams. Gouranga Gupta (talk) 15:29, 27 April 2009 (UTC)
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## colouring

Yesterday I assumed that a sphere could obviously be coloured with 4 colours. Then I realized I didn't actually know this, and it wasn't obvious. Can anyone confirm it by adding a bit to the article?

Maybe you mean the Four color theorem? --Abdull 09:29, 30 May 2006 (UTC)
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## Removal of "Jade Sphere" Graphic

It seems to me that this graphic is irrelevent and adds nothing to the article. It seems more suited to an article concerning digital graphics/photoshop/layers/aqua. Therefore, I am removing it from the sphere article.

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## Differential equation

A sphere of any radius centered at the origin is described by the following differential equation:

$x \, dx + y \, dy + z \, dz = 0.$

What is a sphere of any radius? --Abdull 09:46, 30 May 2006 (UTC)

What is meant is all spheres, regardless of radius satisfy the equation if they are centered at the origin. So
$x^2 + y^2 + z^2 = R^2$
satisfies the above equation, for all R. Sverdrup❞ 10:37, 30 May 2006 (UTC)
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## Proofs

Shouldn't there be links to proofs of claims made on math pages? For instance, the claim that the sphere is the smallest shape enclosing a particular volume may seem obvious, but it would be good to see a proof.

Also, I added the word "locally" to the claim that "surface tension minimizes surface area", since there are actually examples of surface tension creating locally minimum, but not globally minimum, surface area. Flarity 06:33, 22 July 2006 (UTC) its all so easy

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## other usage

does S^n only mean a N sphere or does it have any other meaning? thanks --I got scammed 09:12, 6 September 2006 (UTC)

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## spherical triangle picture

i couldn't figure out how to take the picture out temporarily but, the spherical triangle picture is incorrect. see the discussion of the image.

I've moved it here. --agr 13:49, 29 September 2006 (UTC)
A spherical triangle (red).
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## 2-sphere as 3-dim object

I think it's unnecessary to say that the 2-sphere is a 3-dimensional object. Given the logic (ie. that it can be embedded in R3), a loop, a point, etc. are 3-dimensional objects. But they are also 4, 5, and 6(etc)-dimensional objects as they can be embedded there as well.

Sorry, a 2-sphere is (somewhat non-intuitively) a 2-dimensional object. Tparameter (talk) 04:02, 14 December 2007 (UTC)
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## Removed vandalism

just removed some vandalism from the page (says "removed filth" in the edit log). I am not sure as to whether anything was deleted to put it in and I am not sure how to check. It would be helpful for someone to either do this for me or point me in the direction of instructions, so I know what to do in future. I just had the immediate response of removing the vandalism at the time. Aphswarrior 23:15, 10 January 2007 (UTC)

You got it right - you can check by looking at the page history. For help on reverting vandalism (or anything else that needs reverting), see Help:Reverting. --Zundark 09:27, 11 January 2007 (UTC)
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## Number of points to define a sphere?

What is the minimum number of point needed to define the surface of a unique sphere? Such as 2 points define a line, 3 points define a plane and a unique circle, my guess is that you need at least a fourth point not in the plane of the other three, but are more required?66.202.7.218 14:28, 30 March 2007 (UTC)

Yep. One could think of a one parameter family of spheres which each contain the circle, a fourth point would define one memer of the family. Its a strangly high number consider as you only need 4 numbers center and radius to uniquely define a sphere. --Salix alba (talk) 22:03, 30 March 2007 (UTC)
You got to mind the difference between finite and infinite objects. Any point can be represented by a vector from the origin to the coordinate in question. As you said, a line requires two vectors to define, a plane three, etc. However, in the strictest sense of the terms, lines and planes are infinite objects. A line has no ends, while a plane has no borders. A sphere on the other hand is a finite volume, specifically defined to be 3-dimensional. As for the minimum number of vectors/points required to define it, the simple answer is two (this answer applies to lower and higher dimensional objects of the same family as well). The first vector defines the origin/center of the sphere (let's call it vector 'O'), while the second vector may rest on any arbitrary point on the surface of the sphere (Call it 'P'). By Subtracting O from P, and taking the magnitude (result is distance between surface and center), we get the radius of the sphere. But because this magnitude is constant for any P that lies on the surface (by the very definition of sphere), this vector is not actually required. Instead, a Sphere can be defined using only a vector/point to define the origin, as well as a single scalar to define the radius; hence the final answer is a vector and a scalar. Ghostwo 00:44, 13 December 2007 (UTC)
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## Formula

The article states the following formula under the paragraph equitations, Cartesian coordinate system :

$x = x_0 + r \sin \theta \; \sin \phi$

$y = y_0 + r \cos \theta \; \sin \phi$

Could it be that the formula for x and y are incorrectly reversed ? Both the wiki-page http://en.wikipedia.org/wiki/Spherical_coordinates and http://mathworld.wolfram.com/Sphere.html mention the formula for x and y the other way around. As I am not a mathematician, I post it here so that somebody can verify it. Reaver121 19:56, 14 May 2007 (UTC)

According to Multivariable Calculus 5e p. 1136 (By James Stewart, Published by Thompson Learning), the parametric definition of a sphere in cartesian space is:
$x = x_0 + r \sin \phi \; \cos \theta$
$y = y_0 + r \sin \phi \; \sin \theta$
$z = z_0 + r \cos \phi \;$
Thus, the article seems to be correct as of this writing.
Ghostwo 01:36, 13 December 2007 (UTC)
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## Straight line

A straight line may be defined as the locus of a point which moves radially from another point traversing as the circumference of the circle infinitely,when viewed perpendicularly. —The preceding unsigned comment was added by 59.91.235.11 (talkcontribs) 11 June 07.

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## Change correct?

Is the change at 2007-07-20 07:33 by 194.171.252.100 Section→Equations alright? 194.171.252.100 = Philips Campus-ICT, Eindhoven, NL Maybe there's support here: http://www.math.niu.edu/~rusin/known-math/95/distance Electron9 08:04, 20 July 2007 (UTC)

You can parametrise a sphere in this way, but its not the standard spherical coordinates. I've reverted the change. --Salix alba (talk) 08:29, 20 July 2007 (UTC)

For reference: in mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance from a fixed origin, the elevation angle of that point from a fixed plane, and the azimuth angle of its orthogonal projection on that plane, from a fixed direction on the same. The elevation angle is often replaced by the inclination angle measured from the zenith, the direction perpendicular to the reference plane.--Wikamacallit (talk) 13:19, 26 November 2009 (UTC)

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## London

http://de.wikipedia.org/wiki/Gro%C3%9Fkreis
http://de.wikipedia.org/wiki/Sph%C3%A4rische_Geometrie
http://de.wikipedia.org/wiki/Kugel
http://en.wikipedia.org/wiki/Sphere
http://en.wikipedia.org/wiki/Spherical_geometry

Gegeben:

• 1. Kugel mit einem Radius von 6.000.000 Meter.
• 2. Der Radius ist zugleich die Höhe zum Nordpol und zum Suedpol.
• 3. Der Nordpol und der Suedpol sind Punkte und haben keine Fläche.
• 5. Die gesamte Kugel dreht sich in 24 Stunden einmal um sich selbst.
• 6. Die gesamte Kugel dreht sich bei 365/366 Sonnenscheindauern einmal um den Gasplaneten Sonne im Sonnesystem.
• 7. Der sogenannte Äquator sind 40.075.000 Meter.
• 8. Jeder Punkt zum Nordpol auf der Kugeloberfläche hat eine andere Höhe in Bezug zum Äquatordurchmesser. Der Mount Everest liegt zum Beispiel geschätzt bei der Höhe 3.000.000 * Meter. Der Punkt London liegt zum Beispiel geschätzt in der Höhe von 4.500.000 Meter.
• 9. Der Punkt London ist gegeben.

Gesucht:

• 1. Die Höhe jeden Punktes auf der Geraden vom Mittelpunkt der Kugel zum Nordpol (Bsp. London).
• 2. Die Entfernung vom Mittelpunkt der Kugel zum Punkt London auf der Geraden „dreht sich in 24 Stunden einmal um sich selbst“.
• 3. Eine Art von Berechnung der Fläche von London auf der Kugeloberfläche. —Preceding unsigned comment added by 62.237.32.178 (talk) 13:28, 12 December 2007 (UTC)

Given:

    * 1 Ball with a radius of 6,000,000 meters.
* 2 The radius is also the height to the North Pole and Suedpol.
* 3 The North Pole and the Suedpol are points and have no surface.
* 4 The North Pole and the Suedpol have the numbers 180 degrees degrees and 0 degrees.
* 5 The whole ball rotates in 24 hours by himself
* 6 The whole ball rotates at 365/366 Sonnenscheindauern once around the sun in the solar gas system.
* 7 The so-called Equator are 40,075,000 meters.
* 8 Each point to the North Pole on the ball surface has a different height in relation to the equator diameter. The Mount Everest, for example, is estimated at the height 3,000,000 * meters. The point is, for example, London estimated in the amount of 4,500,000 meters.
* 9 The point is London.


Wanted:

    * 1 The amount of each point on the straight line from the center of the ball to the North Pole (eg London).
* 2 The distance from the center of the ball to the point on the London straights "turns into 24 hours to herself".
* 3 A kind of calculating the surface area of London on the ball's surface.


--Salix alba (talk) 14:54, 12 December 2007 (UTC)

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## Hilbert Space

"[A] sphere is the set of all points in three-dimensional space (R3) which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere"? I'm sorry, but Merriam-Webster disagrees. A sphere is "a solid that is bounded by a surface consisting of all points at a given distance from a point constituting its center". Therefore, a sphere is not bounded by dimension, and, therefore, can only exist in infinite-dimensional [Hilbert space]. This article would be better moved to [3-sphere]. Junulo (talk) 18:13, 16 December 2007 (UTC)

This is about the mathematical entity, not the layperson's definition. That might be at Wiktionary. In either case, I'd say that comparing encylopedia articles with definitions is an apples and oranges endeavor. Tparameter (talk) 14:37, 17 December 2007 (UTC)
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## Surface Area is the differential of Volume

In a maths lesson I noticed that the equation for the Surface Area of a sphere is the differential of the Volume equation. This is not mentioned in the article but I assume this is not just a coincidence, could someone please try to explain this and mabe it needs adding to the article.

$V = \frac{4}{3} \pi r^3$

$\frac{dV}{dr} = 4 \pi r^2$
Mrpowers999 (talk) 13:56, 9 April 2008 (UTC)

Perhaps the inverse phenomena might make more senses: volume is the integral of surface area. Consider a solid sphere, of radius R, made out of series of concentric hollow spheres, Russian-dole style. If the hollow spheres fitted perfectly the volume of the solid sphere would be the sum of the volumns of the hollow spheres. Now the volume of each solid sphere is to the first approximation its surface area times it thickness δr, that is for a hollow sphere of radius r it volume will be $4 \pi r^2 \delta r$ and the sum will be $\sum_{r=0}^R 4 \pi r^2 \delta r$. In the limit you get the integral $\int_{r=0}^R 4 \pi r^2 dr=\frac{4}{3} \pi R^3$. You get a similar result for circumference and area of a circle. --Salix alba (talk) 17:04, 9 April 2008 (UTC)
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## Moved formula of maximal inscribed cube

This edit feels like the solution to a homework exercise, rather than a fact about spheres which passes the basic WP:WEIGHT requirement for inclusion. I have reverted this edit. The text was:

Formula
For the largest cube possibe with all vertices tagent to the surface of the sphere.
2[√⅓(r2)]= L
r = The radius of the sphere.
L = One side of the insribed cube.

This formula is completely out of place without a wider context, and Wikipedia is not an indiscrimate collection of information. (Moreover, I should also mention the side issue that it is unintelligible to talk about a vertex being tangent to a sphere.) siℓℓy rabbit (talk) 00:43, 28 November 2008 (UTC)

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## Formula for volume and area

PLEASE, JUST PUT THE FORMULA FOR THE VOLUME AND AREA OF A SPHERE WHERE IT'S EASY TO SEE !!!! 75.68.200.190 (talk) 15:58, 21 January 2009 (UTC)

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## Request for some simple mathincluding

echoing and adding to above section, no derivations or wierd symbols or wierd fonts, just simple list of Volume =

Surface area =

Circumference

Circumference at a distance r from the equator (eg, the circumference at a given latitude) I think the answer to this is 2(pi)R*cosine(L); L in degrees. It appears that excel 2003 flunks simple math cause the formula =COS(RADIANS(90)) (excel calculates in radians) returns ~~1e-17 where L is the latitude, a value that runs from 0 at the equator to 90o at the pole

how many spheres of diam D1 can pack on a sphere of D2, D1<D2 thanks, Cinnamon colbert (talk) 12:59, 15 May 2009 (UTC)

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## rationale for simple math section

I would say to the math inclined, it is almost impossible to overestimate how intimidated and confused people get by math; a simple section like this is needed, so people don't get lost in the longer sections. —Preceding unsigned comment added by Cinnamon colbert (talkcontribs) 13:21, 17 May 2009 (UTC)

Both formula are at the top of their relevant sections so should be easy enough to find. You addition also had some errors, pi is closer to 3.14 than 3.12 and the volume of a sphere is closer to 4 r^3 than 3 r^3.--Salix (talk): 14:13, 17 May 2009 (UTC)

Dear Salix:

I postulate that the article, as currently written is confusing, and that the simple math section will help a lot of people. The questions then are, (a) is it in fact confusing, and, if so, to how many people, and (b) how do we write the article correctly. I'm not sure how we resolve these questions; I invite your comments. Cinnamon colbert (talk) 12:47, 18 May 2009 (UTC) PS: if you have the formula for packing small spheres on a large sphere, I could reallly use that.

written by www.nu.edu.pk —Preceding unsigned comment added by 121.52.144.12 (talk) 07:57, 22 August 2009 (UTC)

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## Complete proof of sphere value and surface area

Sphere area proof
Half sphera value dividing into 10 discs of radius r=1 and height h=r/10=1/10. From Pythagoras theorem $x=\cos(A)=\sqrt{1-\sin^2(A)}=\sqrt{1-y^2}.$ And so each of ten discs radius: $r_1=r$, $r_2=\sqrt{1-y_2^2}=\sqrt{1-0.1^2}$, $r_3=\sqrt{1-y_3^2}=\sqrt{1-0.2^2}$, $r_4=\sqrt{1-y_4^2}=\sqrt{1-0.3^2}$, ..., $r_{10}=\sqrt{1-y_{10}^2}=\sqrt{1-0.1^2}$.
$V_1=h\pi r^2=\pi r^2 /10=\pi/10=0.1\pi$,
$V_2={\pi\over 10}\cdot (\sqrt{1-0.1^2})^2={\pi(1-0.01)\over 10}=\pi 0.99/10=0.099\pi$,
$V_3=h\pi(1-0.2^2)=\pi(1-0.04)/10=0.96\pi/10=0.096\pi$,
$V_4=h\pi(1-0.3^2)=\pi(1-0.09)/10=0.91\pi/10=0.091\pi$,
$V_5=h\pi(1-0.4^2)=\pi(1-0.16)/10=0.84\pi/10=0.084\pi$,
$V_6=h\pi(1-0.5^2)=\pi(1-0.25)/10=0.75\pi/10=0.075\pi$,
$V_7=h\pi(1-0.6^2)=\pi(1-0.36)/10=0.64\pi/10=0.064\pi$,
$V_8=h\pi(1-0.7^2)=\pi(1-0.49)/10=0.51\pi/10=0.051\pi$,
$V_9=h\pi(1-0.8^2)=\pi(1-0.64)/10=0.36\pi/10=0.036\pi$,
$V_{10}=h\pi(1-0.9^2)=\pi(1-0.81)/10=0.19\pi/10=0.019\pi$.

All sphere value:

$V=2(V_1+V_2+V_3+V_4+V_5+V_6+V_7+V_8+V_9+V_{10})=2\pi(0.1+0.099+0.096+0.091+0.084+0.075+0.064+0.051+0.036+0.019)=2 \pi 0.715=1.43\pi=4.49$.

Close to integrated result:

$V={4\over 3}\pi r^3={4\pi\over 3}\cdot 1^3=4.18879.$
Dividing into more peases gives more precise answer.

Surface area proof
First take a look here.
Each pyramid value is $V={Sh\over 3}$. And sphere can be divided into many such pyramids. And if h=r=1, then surface of sphere will be sphere value multiplied with 1/3. So surface area $4\pi r^2$. —Preceding unsigned comment added by Matematikas1 (talkcontribs) 09:06, 30 December 2009 (UTC)
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## Neutron stars

"An image of one of the most accurate man-made spheres, as it refracts the image of Einstein in the background. This sphere was a fused quartz gyroscope for the Gravity Probe B experiment, and differs in shape from a perfect sphere by no more than 40 atoms of thickness. It is thought that only neutron stars are smoother." This is contradicted by the sourced information at Neutron star "Below the atmosphere one encounters a solid "crust". This crust is extremely hard and very smooth (with maximum surface irregularities of ~5 mm), because of the extreme gravitational field." Kernow (talk) 03:27, 14 May 2010 (UTC)

This is not a direct contradiction. The measure of smoothness of a sphere is relative to its diameter. 5mm in 20km is a smoothness of 4e6. The gyro is reportedly 10nm in 4cm, a smoothness of ... 4e6! GreenAsJade (talk) 11:18, 1 June 2010 (UTC)

This is my first Wikipedia comment, so sorry if I didn't get the format right. Anyway, in the description for the image, it reads "It is thought that only non-rotating neutron stars are smoother." I think this is a somewhat irrational statement because:

a) Celestial objects rotate. b) Neutron stars, as I understand them, rotate at an extremely high angular velocity due to the conservation of angular momentum, since they used to be larger stars.

This statement already has a citation needed. Does anybody have any arguments for keeping it? — Preceding unsigned comment added by 24.22.234.109 (talk) 03:51, 18 November 2011 (UTC)

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## Physics "definition" of a sphere

The Article currently says

"As defined in physics, a sphere is an object (usually idealized for the sake of simplicity) capable of colliding or stacking with other objects which occupy space"

This seems to be failing to mean what it intendes to mean :)

A car, in physics, is capable of colliding with other objects that occupy space. So is the Earth. Neither is taken to be a sphere in physics...

GreenAsJade (talk) 11:21, 1 June 2010 (UTC)

Actually, the Earth is considered to be a sphere... but you're right that this statement needs some clarification.—Tetracube (talk) 16:55, 1 June 2010 (UTC)
It depends on what level of detail you are looking. The Earth is an Oblate Spheroid. I agree it may be a bad example of what is not a sphere though :)
I wanted to pick an entity that more typically under consideration than a car, in physics.

GreenAsJade (talk) 21:39, 1 June 2010 (UTC)

The sentence was introduced with this revision [1] as User:Phasechange's 3rd edit. His edit to Disk (mathematics)
In theoretical physics a disk is a rigid body which is capable of participating in collisions in a two-dimensional gas. Usually the disk is considered rigid so that collisions are deemed elastic.
is slightly better. It seems that he is trying to describe the properties of a sphere to be useful for physics rather than define a physical sphere. A small rewording should clarify the intended meaning. --Salix (talk): 18:10, 1 June 2010 (UTC)

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## area element

The article gives "the area element on the sphere" only "in spherical coordinates", can we also list this in x,y,z??? Jackzhp (talk) 23:32, 18 March 2011 (UTC)

In Cartesian x, y, z coordinates the volume element is simply dx dy dz but the bounds for the integration are quite awkward. An area element in cartesian coordinates could be found by slicing the sphere into thin bands vertically and then slicing in another direction, formula for the resulting segments would be rather awkward. For the sphere it is much easier to work using spherical coordinates which have the nicest formula.--Salix (talk): 00:25, 19 March 2011 (UTC)
The "volume" (i.e., area) form of the sphere can easily be written in Cartesian coordinates. It's just $*dr = \frac{x\,dy\wedge dz + y\,dz\wedge dx + z\, dx\wedge dy}{\sqrt{x^2+y^2+z^2}}$. However, correctly interpreting this expression is probably beyond the scope of the article. Sławomir Biały (talk) 15:19, 28 March 2011 (UTC)
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## Archimedes

It is well known that Archimedes proved that the area of a sphere equals that of a right cylinder of the same radius and height. However, I can't find any evidence that he proved it using an "area-preserving" transformation. So I've added a "citation needed" tag. Can anybody provide a reference? I'd be very interested to read it! best, Sam nead (talk) 02:53, 17 March 2013 (UTC)

Note that, by reference, I mean a reference to one of the works of Archimedes, or to a reliable historian. For example, Serge Tabachnikov says exactly this in a few places, but he doesn't give a citation... Sam nead (talk) 03:03, 17 March 2013 (UTC)

Well, this page http://www.mathpages.com/home/kmath343/kmath343.htm suggests that I may have to eat my words. Still no references, however... Sam nead (talk) 03:15, 17 March 2013 (UTC)

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