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WikiProject Mathematics (Rated Start-class, Mid-priority)
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 Field: Geometry


Do someone know with which software the images in this article were created? GNU

Mathematica, I believe. ✈ James C. 05:15, 2004 Aug 22 (UTC)

Confusion over terms[edit]

The formula for the volume of a spheroid refers to a and b being the axes (a and b usually represent the semi-axes). The formula for the surface area also refers to a and b. In the formula for the volume, do a and b represent the axes or the semi-axes? In the formula for the surface area, do a and b represent the axes or the semi-axes?

Try a = b. Charles Matthews 16:50, 4 May 2005 (UTC)

minor axis / major axis[edit]

I removed this sentence:

"A prolate spheroid has a semi-minor axis shorter than the semi-major axis (a > b); an oblate spheroid has a semi-minor axis longer than the semi-major axis (a < b) and can resemble a disk."

1. Because it's redundant (oblate and prolate are explained above)

2. Because it's completey wrong

The major axis is longer than the minor axis by definition - see ellipse.
Additionally, this sentence leads to the fact, that in the volume calculation always the shorter axis will be squared. This is makes absolutely no sense, it is always the axis not being the rotation axis which will be squared, as it exists in two directions - see ellipsoid

--JogyB 15:08, 12 July 2006 (UTC) (sorry for my english, i'm no native speaker)

The formula and the images[edit]

Unless I'm mistaken, the formula given is for a spheroid with the x-axis of the Cartesian coordinate system as the symmetry axis. The images, however, seem to have the z-axis as the symmetry axis. Shouldn't we change the formula accordingly to


See also Oblate Spheroid and Prolate Spheroid at MathWorld. Or did I go wrong somewhere? Lupo 07:49, 11 October 2005 (UTC)

Prolate spheroid = rugby ball, oblate spheroid = Earth's shape (a little flattened at North and South poles). Charles Matthews 10:13, 11 October 2005 (UTC)

Yes, I knew that, but it doesn't answer my question above. Lupo 11:28, 11 October 2005 (UTC)
I fixed it.--Patrick 13:16, 11 October 2005 (UTC)
I think it is still wrong. Earth is an oblate spheroid: a = the equatorial radius/semi-major axis and b = the north polar radius/semi-minor axis and the south polar radius/semi-minor axis, thus the equation was right as it was
AND, therefore, I believe it should really be "an oblate spheroid has the semi-major axis longer than the two semi-minor, (a > b), and can resemble a disk; a prolate spheroid has the semi-major axis shorter than the two semi-minor (a < b)" (though I did forgot to transfer the "and can resemble a disk" remark).
As for MathWorld, I think it leaves much to be desired! P=/
Thus--unless I'm reading it wrong--the ellipsoid page is also really screwed up. P=( ~Kaimbridge~ 14:14, 11 October 2005 (UTC)
No, x and y play the same role and can be interchanged, they are both divided by the same equatorial half-diameter a, for the Earth larger than the one polar half-diameter b (for only one of the three axes we have this smaller number).--Patrick 15:19, 11 October 2005 (UTC)
Are you sure about that? As I understand it--and I'll be the first to admit I quite easily get lost in any abstract analysis P=)--this is the relevant diagram and equation I understand (realizing that the picture is of an ellipse, not an ellipsoid--though I believe the same oblate/prolate effect applies): If a > b, then you have an oblate ellipsoid; if b > a, then it is prolate; the equation is three terms, to allow different values of b ("b_{north}" and "b_{south}")--? ~Kaimbridge~ 16:45, 11 October 2005 (UTC)
No, you have three, because you have 2 directions for a.--Patrick 20:46, 11 October 2005 (UTC)

The formula for Surface Area also suggests an imaginary value when the spheroid is oblate! (because the eccentricity would be a negative square root).

surface area of spheroids[edit]

Can somebody please check the validity of the following formulae for the surface area of spheroids??

Prolate spheroid ( a > b ):

A = { 2πab2/ (a2 – b2 )1/ 2 }ln{(a + (a2 – b2 )1/ 2 ) / ( a - (a2 – b2 )1/ 2 )}

Oblate spheroid (a < b) :

A = { 4πab2/ (b2 – a2 )1/ 2 }arctan {( (b2 – a2 )1/ 2 ) / a}


Is there a reference for the surface area formulas? I would like to verify them.

Nathalie —Preceding unsigned comment added by (talk) 14:38, 10 February 2010 (UTC)

Planet Earth as an example[edit]

I don't think Earth is a good example. While it is technically correct, most people think of it as a sphere, and it looks like a sphere to the naked eye. To ensure that everyone gets the point immediately, the article needs an example of something that is universally known for being a bit squashed. Nothing comes to mind at the moment unfortunately. Piccadilly 14:19, 6 August 2007 (UTC)

An M&M would work Dr d12 03:01, 23 September 2007 (UTC)

You can also use a pumpkin Eny 21 March 2008 —Preceding unsigned comment added by (talk) 00:01, 22 March 2008 (UTC)

something is wrong here[edit]

A prolate spheroid has a>b, right? But then oe=arccos(a/b) is imaginary? —Preceding unsigned comment added by (talk) 14:05, 29 April 2009 (UTC)

No, with a prolate spheroid, a<b——a>b on an oblate spheroid, like Earth. ~Kaimbridge~ (talk) 14:28, 29 April 2009 (UTC)


I guess "blinded researchers" probably means like "double blind", not with blindfolds on! Someone more familiar with experimental protocol should clarify this section. —Preceding unsigned comment added by (talk) 12:17, 14 October 2009 (UTC)

Volume is indeed given by a single formula - check def of a and b[edit]

Another editor replacd the formula for the volume by this paragraph:

When the spheroid in question is oblate, the volume is \frac{4}{3}\pi a^2b , where a represents the major axis of the ellipse which, when rotated about its minor axis, b, produces the oblate spheroid. When the spheroid in question is prolate, the spheroid is produced by the rotation of an ellipse about it's major axis, hence the volume formula becomes \frac{4}{3}\pi b^2a , where b represents the minor axis of the ellipse which, when rotated about its major axis, produces the prolate spheroid. Hence, the volume of the oblate spheroid which results from the rotation of an ellipse about it's minor axis is always greater than the volume of the prolate spheroid which results from that same ellipse rotated about it's major axis; this is the case whenever a represents the major axis and does not equal b, which represents the minor axis.

This paragraph assumes that the parameters a and b are the major and minor semi-diameters of the ellipse which is the vertical section. However, in the formula given earlier in the article, a and b are the equatorial radius and the polar radius. With this nomenclature, indeed the formula is always (4π/3)a2b for any kind of spheroid. Incidentally this nomenclature is more sensible because it also gives *one* equation (implicit or parametric) for the surface, *one* equation for the area, and so on. All the best, --Jorge Stolfi (talk) 01:07, 28 December 2009 (UTC)

Tumor volume example[edit]

I moved the following example to grading (tumors) since it was far too specialized:

In experimental biology, tumor growth is approximated to take the shape of a spheroid. Often, cancer studies involve the implantation of tumors subcutaneously in mice. Such studies require a simple mechanism by which to evaluate tumor burden. One such method is for two blinded researchers to measure tumor dimensions length and width with calipers. The depth is not measured. Tumor volume in cubic millimeters can be approximated with the following formula: Volume = 0.52(Width^2)Length<ref>W. Su & Q. Wang: Inhibition of Human Prostate Cancer Growth and Prevention of Metastasis Development by Antiangiogenic Activities of Pigment Epithelium-Derived Factor. The Internet Journal of Oncology, 2007 Volume 4 Number 1</ref>

Moreover, in this layman's opinion that method is extremely crude (as it does not measure depth and assumes that the shape is a spheroid). However I retained the approximate formulas, in terms of radii and in termd of diametes (as in the example above). --Jorge Stolfi (talk) 02:25, 28 December 2009 (UTC)

Revised formulae[edit]

The above discussion bears witness to much confusion. If the discussion related to an oblate spheroid only it would make sense to use a,b on the spheroid as conforming to the 'standard' notation for the ellipse. For the prolate case however major and minor are reversed. I have changed notation to agree with current texts which consider both types of spheroid: in particular I agree with the Mathworld discussion. Typos permitting this article now agrees with mathworld. I have removed 'angular eccentricty' since the concept is not used at all in the current literature of mathematics, geodesy, map-projections (according to academic colleagues). The expressions are now written in conventional form.  Peter Mercator (talk) 22:55, 7 January 2012 (UTC)

Question Here...[edit]

Yeah, I don't know the freaking elite math grammar, so is tanh-1 e "tahn(-1)*e" or is it some other concept entirely? Because I'm getting improbably small surface areas by assuming the former. — Preceding unsigned comment added by (talk) 23:35, 28 April 2014 (UTC)

Use \textstyle \tanh^{-1}e = \operatorname{artanh}\, e = \tfrac12\ln\left(\frac{1+e}{1-e}\right), from the article on inverse hyperbolic functions. cffk (talk) 10:01, 29 April 2014 (UTC)

Merger proposal[edit]

I propose Oblate spheroid and Prolate spheroid be merged into this page, and turned into redirects. Much of the content is redundant except for the real-world examples of Oblate/Prolate spheroids. (talk) 01:48, 8 June 2014 (UTC)

I think this is a good idea. cffk (talk) 03:29, 8 June 2014 (UTC)
I agree. Fgnievinski (talk) 00:05, 13 August 2014 (UTC)
Great idea. +mt 22:40, 1 September 2014 (UTC)
I agree. Bharat Singhvi 12:48, 5 September 2014 (UTC)
Agreed. Jaywilson (talk) 21:15, 17 November 2014 (UTC)
Yes BUT. Why stop there? The article Ellipsoid also covers the same material. I suggest that the three spheroid articles be merged into Ellipsoid using the figures therein and dropping some pretty pictures and possibly some of the uses/applications. We also need to include a discussion of nomenclature in the article, ie usage of spheroid/ ellipsoid of revolution/ ellipsoid as a function of time and subject area. Who will volunteer? — Peter Mercator (talk) 14:18, 18 November 2014 (UTC)
Disagree; in fact, portions of ellipsoid detailing ellipsoid of revolution would better be moved into spheroid. Fgnievinski (talk) 18:45, 18 November 2014 (UTC)