# Talk:Pyramid (geometry)

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I'm new to wikipedia so I wanted to check if adding a section about the formulas for finding the surface area of pyramids would be all right. Because I think it would be fine. --Learning is good 04:32, 26 November 2006 (UTC)

• Added a surface area formula MathsIsFun 05:04, 26 November 2006 (UTC)

Thanks --Learning is good 05:12, 26 November 2006 (UTC)

"Since the area of any shape is multiplied by the square of the shape's scaling factor, the area of a cross section at height y is ..." Is this confusing to anyone else? Is there a work missing like "Since the area of any shape is <the base> multiplied by the square of the shape's scaling factor...>. I know <the base> is not the right word, but I don't understand the sentence as it is. --Rusty Maths —Preceding unsigned comment added by 84.203.137.22 (talk) 11:00, 11 January 2008 (UTC)

## Volume and surface area formulæ

We need to stop edit warring on the volume and surface area sections. Here are issues that need to be addressed:

• We need to find a reliable citations for these formula.
• Proofs and derivations are not really necessary unless they are concise and to-the-point. These should be replaced with a citation instead of cluttering the page.
• We should not post formulæ derived from our own calculations, based on parameters that are not general for all pyramids, or worse yet, based on assumptions that aren't always true.

Tetracube (talk) 22:38, 26 March 2009 (UTC)

#### ==

Another poorly written article from Wikipedia. So there is someone, apparently more than one person, who believes a pyramid has a radius. I am certainly no expert, but I believe this to be false. If it is true it, it is certainly NOT something that is known broadly enough to be used without explanation. Perhaps the authors meant the circle on which the vertices of the base lie? If so, this would only be true for those with a regular base, right? The article needs to clearly explicitly state when it referring to general prisms and when it is about square pyramids. This should happen in every section. Why is this not self-evident?71.31.149.105 (talk) 23:23, 30 March 2012 (UTC)

## MOSMATH

Please: WP:MOSMATH exists. Don't write things like n+1. The correct notation is n + 1 (the n is italicized; the 1 is not; proper spacing is used). Michael Hardy (talk) 13:23, 28 March 2009 (UTC)

## Johnsons

This language: "Besides the triangular pyramid, only the square and pentagonal pyramids can be composed of regular convex polygons, in which case they are Johnson solids." is not correct. A Johnson Solid is any convex polyhedrod... so a pyramid, which is a conical solid with a polygon base of n size, will always be convex. If the statement is true, and for some reason I'm mis-reading the information about Johnson solids, then the above-quoted statement needs to be referenced/cited. Alphachimera (talk) 17:00, 5 August 2009 (UTC)

A Johnson solid is a convex nonuniform solid with regular faces. You can't make a pyramid with a regular heptagonal base and regular (equilateral) triangle sides, because the distance from the corners to the center of the heptagon is longer than the side. —Tamfang (talk) 05:17, 6 August 2009 (UTC)

## Six pyramids forming a cube

I deleted the paragraph stating that the factor of 1/3 can be derived by the fact that a cube can be divided into 6 equal pyramids. The converse is not true; the only pyramid that can form a cube in this way has a side-to-base dihedral angle of precisely 45° and a base that is exactly square. Other pyramids cannot form a cube in this way. Even if you allow distortion of a cube into cuboidal shapes, the constituent pyramids will not be equal, and therefore you cannot derive the 1/3 factor from it in any trivial way. In other words, this intuition is completely crap. Why not just stick with the real mathematical derivation of the formula in the first place?—Tetracube (talk) 17:24, 21 September 2009 (UTC)

Because for a lot of people who are less mathematical, it's easier to understand. Sometimes what Ian Stewart and Terry Pratchett called lies-to-children (see Science of Discworld) may be incomplete or entirely false, and yet still useful. However, the main volume paragraph has been rewritten to be a little clearer since I added that (now deleted) paragraph, after someone had come to me and said they found the current page utterly incomprehensible. Barefootliam (talk) 05:17, 10 February 2010 (UTC)

## Simple proof of piramide value

Divide piramide into 10 parts and each part of height c=1 and so h=1*10=10, a=10, ${\displaystyle B=a^{2}=100}$. So aproximate piramide value is:

${\displaystyle V=c(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}+a_{7}^{2}+a_{8}^{2}+a_{10}^{2})=1(1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}+7^{2}+8^{2}+9^{2}+10^{2})=385.}$

And this pretty close to ${\displaystyle {ha^{2} \over 3}={10\cdot 10^{2} \over 3}=333.3333}$.

"pretty close" is not a proof. —Tamfang (talk) 05:45, 31 December 2009 (UTC)
Divide into more peaces and you will get infinitly close to 333.(3). Like for example:
${\displaystyle V=c(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}+a_{7}^{2}+a_{8}^{2}+a_{10}^{2}+...+a_{99}^{2}+a_{100}^{2})=}$

${\displaystyle =0.1(0.1^{2}+0.2^{2}+0.3^{2}+0.4^{2}+0.5^{2}+0.6^{2}+0.7^{2}+0.8^{2}+0.9^{2}+0.1^{2}+1.1^{2}+1.2^{2}+1.3^{2}+1.4^{2}+...+9.8^{2}+9.9^{2}+10^{2})=}$

${\displaystyle =333.520}$.
And if ${\displaystyle h=100}$ and ${\displaystyle a=10}$, then it is also correct:
${\displaystyle V=c(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}+a_{7}^{2}+a_{8}^{2}+a_{10}^{2}+...+a_{99}^{2}+a_{100}^{2})=}$

${\displaystyle =1(0.1^{2}+0.2^{2}+0.3^{2}+0.4^{2}+0.5^{2}+0.6^{2}+0.7^{2}+0.8^{2}+0.9^{2}+0.1^{2}+1.1^{2}+1.2^{2}+1.3^{2}+1.4^{2}+...+9.8^{2}+9.9^{2}+10^{2})=}$

${\displaystyle =3335.20}$.
So like ${\displaystyle V={Bh \over 3}={a^{2}h \over 3}={10^{2}\cdot 100 \over 3}={10000 \over 3}=3333.(3)}$.
It's like integral sums of peaces under parabola ${\displaystyle a=x}$, ${\displaystyle \int _{0}^{10}x^{2}dx={x^{3} \over 3}|_{0}^{10}={10^{3} \over 3}-{0^{3} \over 3}=333.3333}$ - it's answer when ${\displaystyle a=h}$. All peaces under parabola have same width ${\displaystyle c=1}$, but height ${\displaystyle x^{2}}$ of each peace like ${\displaystyle 1^{2}}$, then ${\displaystyle 2^{2}}$, then ${\displaystyle 3^{2}}$,...,${\displaystyle 9^{2}}$, ${\displaystyle 10^{2}}$. So this ${\displaystyle y=x^{2}}$ at any point on x axis, can be interpretated as ${\displaystyle B=a^{2}}$ and this each peace of pyramid height is ${\displaystyle c=1}$ and each peace value ${\displaystyle V_{1}=ca_{1}^{2}}$, ${\displaystyle V_{2}=ca_{2}^{2}}$, ${\displaystyle V_{3}=ca_{3}^{2}}$,..., ${\displaystyle V_{9}=ca_{9}^{2}}$, ${\displaystyle V_{10}=ca_{10}^{2}}$. And ${\displaystyle V=V_{1}+V_{2}+V_{3}+...+V_{9}+V_{10}=385}$.
And if ${\displaystyle a=100}$, ${\displaystyle h=100}$, ${\displaystyle c=1}$, then of course it's also correct:
${\displaystyle V=c(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}+a_{7}^{2}+a_{8}^{2}+a_{10}^{2}+...+a_{99}^{2}+a_{100}^{2})=}$

${\displaystyle =1(1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}+7^{2}+8^{2}+9^{2}+10^{2}+11^{2}+12^{2}+13^{2}+14^{2}+...+98^{2}+99^{2}+100^{2})=333520}$. Very close to ${\displaystyle V={ha^{2} \over 3}={100\cdot 100^{2} \over 3}=333333.(3)}$.

And if ${\displaystyle a=100,}$ ${\displaystyle h=10}$, ${\displaystyle c=0.1}$, dividing pyramid into 100 parts (c=h/100=0.1) and it is correct:
${\displaystyle V=c(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}+a_{7}^{2}+a_{8}^{2}+a_{10}^{2}+...+a_{99}^{2}+a_{100}^{2})=}$

${\displaystyle =0.1(1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}+7^{2}+8^{2}+9^{2}+10^{2}+11^{2}+12^{2}+13^{2}+14^{2}+...+98^{2}+99^{2}+100^{2})=33352}$. Very close to ${\displaystyle V={ha^{2} \over 3}={10\cdot 100^{2} \over 3}=33333.(3)}$.

No matter how awesomely close you get, it's not a proof without some rigorous reason for confidence in the limit. If you understand integration for area, why not simply integrate for volume? Nor is the value for a single case all that interesting. —Tamfang (talk) 05:25, 16 February 2010 (UTC)

## proof of surface area of piramid

Pyramid base is square. Pyramide base ${\displaystyle B=a^{2},}$ a=1, pyramid height is H=1. Now need to find each of 4 triangles area, but then need to know h of triangle, because a=1 of triangle base. So ${\displaystyle h={\sqrt {H^{2}+({a \over 2})^{2}}}={\sqrt {1+{1 \over 4}}}={\sqrt {5 \over 4}}={\sqrt {1.25}}=1.118033989}$.

So total surface area of pyramid:
${\displaystyle S=B+4\cdot {ha \over 2}=1+4\cdot {1.118033989\cdot 1 \over 2}=1+2\cdot 1.118033989=1+2.236067978=3.236067978}$.
This is still only for one particular pyramid. We already have a general formula given in the article (${\displaystyle A=B+{\frac {PL}{2}}}$ where B is the base area, P is the base perimeter and L is the slant height). If you would like to generalise your formula, then it could be added as an alternative, but don't substitute values. Dbfirs 11:43, 11 October 2010 (UTC)

## Since when was a tetrahedron a pyramid?

Just wondering why a tetrahedron, and shapes with more than four sides on their base, are being classed as pyramids.--78.146.175.222 (talk) 12:45, 9 February 2013 (UTC)

Mathematically, any shape with a polygonal base and plane faces sloping to a vertex is called a pyramid, but culturally nearly all pyramids have square bases (there might be rare exceptions somewhere?). The article covers both usages. Should this be clarified? Dbfirs 21:35, 9 February 2013 (UTC)

[1] a. A solid figure with a polygonal base and triangular faces that meet at a common point. Tom Ruen (talk) 22:42, 9 February 2013 (UTC)

## Simplices, complices, duplices?

I notice that the English plural simplexes has been changed to the "false Latin" simplices. I'm not going to revert this because it is probably the case that some mathematicians use this etymologically dubious plural (simplex is not a Latin noun), and this is a mathematical article, but I question the claim in the edit summary that a false Latin plural is "more correct". Dbfirs 07:32, 21 March 2013 (UTC)

## Removed section on symmetry

A grumpy editor removed this section. Tom Ruen (talk) 07:44, 11 June 2015 (UTC)

 Symmetry In general, any planar polygon can represent the base of a pyramid. The symmetry of a right pyramid will be the same as the base polygon. A right pyramid with a regular polygon base with 2D dihedral symmetry Dihn, order 2n has Cnv in Schoenflies_notation.[1] A base polygon with 2D cyclic symmetry Cn, order n is called also Cn in 3-dimensions. An oblique pyramid in general, whether acute, right-angled, or obtuse, has no symmetry, but it can have mirror symmetry if the apex is directly above a mirror line of the base polygon. A triangular based pyramid may also have higher symmetries from other apex-base orientations as well.[citation needed][original research?] For example, on a square pyramid, with the apex colored blue, and vertical height drawn in red, and with blue symmetry lines drawn on the base:
Because it's original research, and (as is likely with original research) partly incorrect. In particular I have twice pointed out that it is not true that the symmetry of a right pyramid is not always the same as the base. The regular tetrahedron is an exception, with more symmetry than its base. —David Eppstein (talk) 14:38, 11 June 2015 (UTC)
You're just being grumpy, expressing a special case as if its a flaw. With the base distinct from the sides, the symmetry is correct. So instead of improving the wording to fit your mental perfection, you remove the whole section. Why not just say you don't want this information up? That would be more honest. Should we also say a general nonsquare rhombus has Dih2 symmetry to make sure people don't get confused that a square rhombus also only has order-4 symmetry? Tom Ruen (talk) 15:20, 11 June 2015 (UTC)
It makes me grumpy when people think they are doing mathematics, get it wrong, present their mistakes to the world on Wikipedia as if they were truths, and then try to defend the mistakes. —David Eppstein (talk) 16:19, 11 June 2015 (UTC)
Yes, indeed, and I just looked up what Coxeter said on the symmetry of pyramids, and imagine that, he also fails to give the regular tetrahedron as a special case as higher symmetry than a general regular right pyramid. He must have been a poor thinker and writer too, just like me. Tom Ruen (talk) 16:24, 11 June 2015 (UTC)
BTW, that is not the only exception. It is also possible for a right pyramid over an iscosceles triangle to have more symmetry than its base. —David Eppstein (talk) 20:12, 12 June 2015 (UTC)
How do you suggest you express these exceptions? Currently it says A triangular based pyramid may also have higher symmetries from other apex-base orientations as well. Tom Ruen (talk) 16:30, 13 June 2015 (UTC)
I would suggest we find a reliable source that gets it right and follow what they do. That is the correct way of avoiding original research. —David Eppstein (talk) 17:12, 13 June 2015 (UTC)
That's the whole problem. You can quote sources without understanding the limitations of those source, or which can also ignore special cases, and you're back where you started. So the only answer is to assume the general case is what we're interested in, not special cases.
And as I already contrasted, do we need to say a Rhombus has Dih2 symmetry, unless its a square, in which case it has Dih4 symmetry. And both are true statements, but what if no sources say that, what if that is synthesis? What if there's a different special case of a rhombus with higher symmetry, like a skew rhombus of course which can have 3D D2d symmetry. What if no sources say that? Maybe we have to add a qualifier planar rhombi. Would that be enough? What if we can't dare say a general rhombus has Dih2 symmetry because an imaginary case we can't see, or which all sources ever written don't say, then we can't say it. And if we say "well, let's just exclude the square and assume no other cases exist.", but no source explicitly says that. It's hopeless by unless you brow beat some "reliable" author to explicitly prove all cases, and states them and fails to make any mistakes and fails to ignore specific cases. Maybe someone should write a PhD thesis on the symmetry of a general rhombus?! Tom Ruen (talk) 17:47, 13 June 2015 (UTC)
• ^ H. S. M. Coxeter, Introduction to Solid Geometry, second edition, 1969, ISBN 72-93903