Square pyramid#Surface area and volume

A square pyramid has five vertices, eight edges, and five faces. One face, called the base of the pyramid, is a square; the four other faces are triangles.{{sfnp|Clissold|2020|p = [https://books.google.com/books?id=XgW5DwAAQBAJ&pg=PA180 180]}} Four of the edges make up the square by connecting its four vertices. The other four edges are known as the lateral edges of the pyramid; they meet at the fifth vertex, called the apex.{{sfnmp

| 1a1 = O'Keeffe | 1a2 = Hyde | 1y = 2020 | 1p = [https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA141 141]

| 2a1 = Smith | 2y= 2000 | 2p = [https://books.google.com/books?id=B0khWEZmOlwC&pg=PA98 98]

}} If the pyramid's apex lies on a line erected perpendicularly from the center of the square, it is called a right square pyramid, and the four triangular faces are isosceles triangles. Otherwise, the pyramid has two or more non-isosceles triangular faces and is called an oblique square pyramid.{{sfnp|Freitag|2014|p=[https://books.google.com/books?id=GYsWAAAAQBAJ&pg=PA598 598]}}

The slant height $s$ of a right square pyramid is defined as the height of one of its isosceles triangles. It can be obtained via the Pythagorean theorem:

$$s\; =\; \backslash sqrt\{b^2\; -\; \backslash frac\{l^2\}\{4\}\},$$

where $l$ is the length of the triangle's base, also one of the square's edges, and $b$ is the length of the triangle's legs, which are lateral edges of the pyramid.{{sfnmp

| 1a1 = Larcombe | 1y = 1929 | 1p = [https://books.google.com/books?id=SAE9AAAAIAAJ&pg=PA177 177]

| 2a1 = Perry | 2a2 = Perry | 2y = 1981 | 2pp = [https://books.google.com/books?id=Di2uCwAAQBAJ&pg=PA145 145–146]

}} The height $h$ of a right square pyramid can be similarly obtained, with a substitution of the slant height formula giving:{{sfnp|Larcombe|1929|p=[https://books.google.com/books?id=SAE9AAAAIAAJ&pg=PA177 177]}}

$$h\; =\; \backslash sqrt\{s^2\; -\; \backslash frac\{l^2\}\{4\}\}\; =\; \backslash sqrt\{b^2\; -\; \backslash frac\{l^2\}\{2\}\}.$$

A polyhedron's surface area is the sum of the areas of its faces. The surface area $A$ of a right square pyramid can be expressed as $A\; =\; 4T\; +\; S$, where $T$ and $S$ are the areas of one of its triangles and its base, respectively. The area of a triangle is half of the product of its base and side, with the area of a square being the length of the side squared. This gives the expression:{{sfnp|Freitag|2014|p=[https://books.google.com/books?id=GYsWAAAAQBAJ&pg=PA798 798]}}

$$A\; =\; 4\backslash left(\backslash frac\{1\}\{2\}ls\backslash right)\; +\; l^2\; =\; 2ls\; +\; l^2.$$

In general, the volume $V$ of a pyramid is equal to one-third of the area of its base multiplied by its height.{{sfnp|Alexander|Koeberlin|2014|p=[https://books.google.com/books?id=EN_KAgAAQBAJ&pg=PA403 403]}} Expressed in a formula for a square pyramid, this is:{{sfnp|Larcombe|1929|p=[https://books.google.com/books?id=SAE9AAAAIAAJ&pg=PA178 178]}}

$$V\; =\; \backslash frac\{1\}\{3\}l^2h.$$

Many mathematicians have discovered the formula for calculating the volume of a square pyramid in ancient times. In the Moscow Mathematical Papyrus, Egyptian mathematicians demonstrated knowledge of the formula for calculating the volume of a truncated square pyramid, suggesting that they were also acquainted with the volume of a square pyramid, but it is unknown how the formula was derived. Beyond the discovery of the volume of a square pyramid, the problem of finding the slope and height of a square pyramid can be found in the Rhind Mathematical Papyrus.{{sfnp|Cromwell|1997|pp=[https://archive.org/details/polyhedra0000crom/page/20/mode/2up?view=theater 20–22]}} The Babylonian mathematicians also considered the volume of a frustum, but gave an incorrect formula for it.{{sfnp|Eves|1997|p=[https://books.google.com/books?id=J9QcmFHj8EwC&pg=PA2 2]}} One Chinese mathematician Liu Hui also discovered the volume by the method of dissecting a rectangular solid into pieces.{{sfnp|Wagner|1979}}

File:J1 square pyramid.stl

{{anchor|Equilateral square pyramid}}If all triangular edges are of equal length, the four triangles are equilateral, and the pyramid's faces are all regular polygons, it is an equilateral square pyramid.{{sfnp|Hocevar|1903|p=[https://books.google.com/books?id=0OAXAAAAYAAJ&pg=PA44 44]}} The dihedral angles between adjacent triangular faces are $\backslash arccos\; \backslash left(-1/3\; \backslash right)\; \backslash approx\; 109.47^\backslash circ$, and that between the base and each triangular face being half of that, $\backslash arctan\; \backslash left(\backslash sqrt\{2\}\backslash right)\; \backslash approx\; 54.74^\backslash circ$.{{sfnp|Johnson|1966}} A convex polyhedron in which all of the faces are regular polygons is called a Johnson solid. The equilateral square pyramid is among them, enumerated as the first Johnson solid $J\_1$.{{sfnp|Uehara|2020|p=[https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62 62]}}

Because its edges are all equal in length (that is, $b\; =\; l$), its slant, height, surface area, and volume can be derived by substituting the formulas of a right square pyramid:{{sfnmp

| 1a1 = Simonson | 1y = 2011 | 1p = [https://books.google.com/books?id=Ws6-DwAAQBAJ&pg=PA123 123]

| 2a1 = Berman | 2y = 1971 | 2loc = see table IV, line 21

}}

\begin{align}

s = \frac{\sqrt{3}}{2}l \approx 0.866l, &\qquad h = \frac{1}{\sqrt{2}}l \approx 0.707l,\\

A = (1 + \sqrt{3})l^2 \approx 2.732l^2, &\qquad V = \frac{\sqrt{2}}{6}l^3 \approx 0.236l^3.

\end{align}

Like other right pyramids with a regular polygon as a base, a right square pyramid has pyramidal symmetry. For the square pyramid, this is the symmetry of cyclic group $C\_\{4\backslash mathrm\{v\}\}$: the pyramid is left invariant by rotations of one-, two-, and three-quarters of a full turn around its axis of symmetry, the line connecting the apex to the center of the base; and is also mirror symmetric relative to any perpendicular plane passing through a bisector of the base.{{sfnp|Johnson|1966}} It can be represented as the wheel graph $W\_4$, meaning its skeleton can be interpreted as a square in which its four vertices connects a vertex in the center called the universal vertex.{{sfnp|Pisanski|Servatius|2013|p=[https://books.google.com/books?id=3vnEcMCx0HkC&pg=PA21 21]}} It is self-dual, meaning its dual polyhedron is the square pyramid itself.{{sfnp|Wohlleben|2019|p=[https://books.google.com/books?id=rEpjDwAAQBAJ&pg=PA485 485–486]}}

An equilateral square pyramid is an elementary polyhedron. This means it cannot be separated by a plane to create two small convex polyhedrons with regular faces.{{sfnmp

| 1a1 = Hartshorne | 1y = 2000 | 1p = [https://books.google.com/books?id=EJCSL9S6la0C&pg=PA464 464]

| 2a1 = Johnson | 2y = 1966

}}

{{multiple image

| total_width = 400

| align = right

| image1 = All Gizah Pyramids.jpg

| caption1 = The Egyptian pyramids are examples of square pyramidal buildings in architecture.

| image2 = Piramide Chichen-Itza - panoramio (2).jpg

| caption2 = One of the Mesoamerican pyramids, a similar building to the Egyptian, has flat tops and stairs at the faces

}}

In architecture, the pyramids built in ancient Egypt are examples of buildings shaped like square pyramids.{{sfnp|Kinsey|Moore|Prassidis|2011|p=[https://books.google.com/books?id=fFpuDwAAQBAJ&pg=RA1-PA371 371]}} Pyramidologists have put forward various suggestions for the design of the Great Pyramid of Giza, including a theory based on the Kepler triangle and the golden ratio. However, modern scholars favor descriptions using integer ratios, as being more consistent with the knowledge of Egyptian mathematics and proportion.{{harvtxt|Herz-Fischler|2000}} surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle. See {{harvtxt|Rossi|2004}}, pp. [https://archive.org/details/architechture-and-mathematics-in-ancient-egypt-corianna-rossi-2003/page/67/ 67–68], quoting that "there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to $\backslash varphi$, and $\backslash varphi$ itself as a number, do not fit with the extant Middle Kingdom mathematical sources"; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56. See also {{harvtxt|Rossi|Tout|2002}} and {{harvtxt|Markowsky|1992}}. The Mesoamerican pyramids are also ancient pyramidal buildings similar to the Egyptian; they differ in having flat tops and stairs ascending their faces.{{sfnmp

| 1a1 = Feder | 1y = 2010 | 1p = [https://books.google.com/books?id=RlRz2symkAsC&pg=PA34 34]

| 2a1 = Takacs | 2a2 = Cline | 2y = 2015 | 2p = [https://books.google.com/books?id=SPcvCgAAQBAJ&pg=PA16 16]

}} Modern buildings whose designs imitate the Egyptian pyramids include the Louvre Pyramid and the casino hotel Luxor Las Vegas.{{sfnmp

| 1a1 = Jarvis | 1a2 = Naested | 1y = 2012 | 1p = [https://books.google.com/books?id=NWzsz8vioZwC&pg=PA172 172]

| 2a1 = Simonson | 2y = 2011 | 2p = [https://books.google.com/books?id=Ws6-DwAAQBAJ&pg=PA122 122]

}}

In stereochemistry, an atom cluster can have a square pyramidal geometry. A square pyramidal molecule has a main-group element with one active lone pair, which can be described by a model that predicts the geometry of molecules known as VSEPR theory.{{sfnp|Petrucci|Harwood|Herring|2002|p=[https://books.google.com/books?id=EZEoAAAAYAAJ&pg=PA414 414]}} Examples of molecules with this structure include chlorine pentafluoride, bromine pentafluoride, and iodine pentafluoride.{{sfnp|Emeléus|1969|p=[https://books.google.com/books?id=9SkSBQAAQBAJ&pg=PA13 13]}}

File:Tetrakishexahedron.jpg, a construction of polyhedra by augmentation involving square pyramids]]

The base of a square pyramid can be attached to a square face of another polyhedron to construct new polyhedra, an example of augmentation. For example, a tetrakis hexahedron can be constructed by attaching the base of an equilateral square pyramid onto each face of a cube.{{sfnp|Demey|Smessaert|2017}} Attaching prisms or antiprisms to pyramids is known as elongation or gyroelongation, respectively.{{sfnp|Slobodan|Obradović|Ðukanović|2015}} Some of the other Johnson solids can be constructed by either augmenting square pyramids or augmenting other shapes with square pyramids: elongated square pyramid $J\_8$, gyroelongated square pyramid $J\_\{10\}$, elongated square bipyramid $J\_\{15\}$, gyroelongated square bipyramid $J\_\{17\}$, augmented triangular prism $J\_\{49\}$, biaugmented triangular prism $J\_\{50\}$, triaugmented triangular prism $J\_\{51\}$, augmented pentagonal prism $J\_\{52\}$, biaugmented pentagonal prism $J\_\{53\}$, augmented hexagonal prism $J\_\{54\}$, parabiaugmented hexagonal prism $J\_\{55\}$, metabiaugmented hexagonal prism $J\_\{56\}$, triaugmented hexagonal prism $J\_\{57\}$, and augmented sphenocorona $J\_\{87\}$.{{harvtxt|Rajwade|2001}}, pp. [https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84 84–89]. See Table 12.3, where $P\_n$ denotes the {{nowrap|$n$-sided}} prism and $A\_n$ denotes the {{nowrap|$n$-sided}} antiprism.

• Square pyramidal number, a natural number that counts the number of stacked spheres in a square pyramid.

{{Clear}}

{{reflist|25em}}

{{refbegin|30em}}

• {{cite book

| last1 = Alexander | first1 = Daniel C.

| last2 = Koeberlin | first2 = Geralyn M.

| year = 2014

| title = Elementary Geometry for College Students

| url = https://books.google.com/books?id=EN_KAgAAQBAJ

| edition = 6th

| publisher = Cengage Learning

| isbn = 978-1-285-19569-8

}}

• {{cite journal

| last = Berman | first = Martin

| year = 1971

| title = Regular-faced convex polyhedra

| journal = Journal of the Franklin Institute

| volume = 291

| issue = 5

| pages = 329–352

| doi = 10.1016/0016-0032(71)90071-8

| mr = 290245

}}

• {{cite book

| last = Clissold | first = Caroline

| year = 2020

| title = Maths 5–11: A Guide for Teachers

| publisher = Taylor & Francis

| url = https://books.google.com/books?id=XgW5DwAAQBAJ

| isbn = 978-0-429-26907-3

}}

• {{cite book

| last = Cromwell | first = Peter R.

| title = Polyhedra

| year = 1997

| url = https://archive.org/details/polyhedra0000crom

| publisher = Cambridge University Press

| isbn = 978-0-521-55432-9

}}

• {{cite journal

| last1 = Demey | first1 = Lorenz

| last2 = Smessaert | first2 = Hans

| title = Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

| journal = Symmetry

| date = 2017

| volume = 9

| issue = 10

| page = 204

| doi = 10.3390/sym9100204

| doi-access = free

| bibcode = 2017Symm....9..204D

}}

• {{cite book

| last = Emeléus | first = H. J. | authorlink = Harry Julius Emeléus

| title = The Chemistry of Fluorine and Its Compounds

| year = 1969

| url = https://books.google.com/books?id=9SkSBQAAQBAJ

| publisher = Academic Press

| isbn = 978-1-4832-7304-4

}}

• {{cite book

| last = Eves | first = Howard | authorlink = Howard Eves

| year = 1997

| title = Foundations and Fundamental Concepts of Mathematics

| edition = 3rd

| url = https://books.google.com/books?id=J9QcmFHj8EwC

| publisher = Dover Publications

| isbn = 978-0-486-69609-6

}}

• {{cite book

| last = Feder | first = Kenneth L.

| year = 2010

| title = Encyclopedia of Dubious Archaeology: From Atlantis to the Walam Olum: From Atlantis to the Walam Olum

| publisher = ABC-CLIO

| url = https://books.google.com/books?id=RlRz2symkAsC

| isbn = 978-0-313-37919-2

}}

• {{cite book

| last = Freitag | first = Mark A.

| year = 2014

| title = Mathematics for Elementary School Teachers: A Process Approach

| publisher = Brooks/Cole

| isbn = 978-0-618-61008-2

| url = https://books.google.com/books?id=GYsWAAAAQBAJ

}}

• {{cite book

| last = Hartshorne | first = Robin | author-link = Robin Hartshorne

| year = 2000

| title = Geometry: Euclid and Beyond

| series = Undergraduate Texts in Mathematics

| publisher = Springer-Verlag

| isbn = 9780387986500

| url = https://books.google.com/books?id=EJCSL9S6la0C

}}

• {{cite book

| last = Herz-Fischler

| first = Roger

| year = 2000

| isbn = 0-88920-324-5

| publisher = Wilfrid Laurier University Press

| title = The Shape of the Great Pyramid

}}

• {{cite book

| last = Hocevar | first = Franx

| year = 1903

| url = https://books.google.com/books?id=0OAXAAAAYAAJ

| title = Solid Geometry

| publisher = A. & C. Black

}}

• {{cite book

| last1 = Jarvis | first1 = Daniel

| last2 = Naested | first2 = Irene

| title = Exploring the Math and Art Connection: Teaching and Learning Between the Lines

| year = 2012

| publisher = Brush Education

| isbn = 978-1-55059-398-3

}}

• {{cite journal

| last = Johnson | first = Norman W. | authorlink = Norman W. Johnson

| year = 1966

| title = Convex polyhedra with regular faces

| journal = Canadian Journal of Mathematics

| volume = 18

| pages = 169–200

| doi = 10.4153/cjm-1966-021-8

| mr = 0185507

| s2cid = 122006114

| zbl = 0132.14603| doi-access = free

}}

• {{cite book

| last1 = Kinsey | first1 = L. Christine | authorlink1 = L. Christine Kinsey

| last2 = Moore | first2 = Teresa E.

| last3 = Prassidis | first3 = Efstratios

| title = Geometry and Symmetry

| year = 2011

| url = https://books.google.com/books?id=fFpuDwAAQBAJ

| publisher = John Wiley & Sons

| isbn = 978-0-470-49949-8

}}

• {{cite book

| last = Larcombe | first = H. J.

| title = Cambridge Intermediate Mathematics: Geometry Part II

| year = 1929

| url = https://books.google.com/books?id=SAE9AAAAIAAJ

| publisher = Cambridge University Press

}}

• {{cite journal

| last = Markowsky | first = George

| year = 1992

| title = Misconceptions about the Golden Ratio

| journal = The College Mathematics Journal

| volume = 23

| issue = 1

| doi = 10.2307/2686193

| url = http://www.umcs.maine.edu/~markov/GoldenRatio.pdf

| publisher = Mathematical Association of America

| pages = 2–19

| jstor = 2686193

| access-date = 29 June 2012

}}

• {{cite book

| last1 = O'Keeffe | first1 = Michael

| last2 = Hyde | first2 = Bruce G.

| title = Crystal Structures: Patterns and Symmetry

| year = 2020

| url = https://books.google.com/books?id=_MjPDwAAQBAJ

| publisher = Dover Publications

| isbn = 978-0-486-83654-6

}}

• {{cite book

| last1 = Perry | first1 = O. W.

| last2 = Perry | first2 = J.

| title = Mathematics

| year = 1981

| url = https://books.google.com/books?id=Di2uCwAAQBAJ

| publisher = Springer

| isbn = 978-1-349-05230-1

| doi = 10.1007/978-1-349-05230-1

}}

• {{cite book

| last1 = Petrucci | first1 = Ralph H.

| last2 = Harwood | first2 = William S.

| last3 = Herring | first3 = F. Geoffrey

| title = General Chemistry: Principles and Modern Applications

| volume = 1

| year = 2002

| url = https://books.google.com/books?id=EZEoAAAAYAAJ

| publisher = Prentice Hall

| isbn = 978-0-13-014329-7

}}

• {{cite book

| last1 = Pisanski | first1 = Tomaž

| last2 = Servatius | first2 = Brigitte

| title = Configuration from a Graphical Viewpoint

| year = 2013

| url = https://books.google.com/books?id=3vnEcMCx0HkC

| publisher = Springer

| isbn = 978-0-8176-8363-4

| doi = 10.1007/978-0-8176-8364-1

}}

• {{cite book

| last = Rajwade | first = A. R.

| title = Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem

| series = Texts and Readings in Mathematics

| year = 2001

| url = https://books.google.com/books?id=afJdDwAAQBAJ

| publisher = Hindustan Book Agency

| isbn = 978-93-86279-06-4

| doi = 10.1007/978-93-86279-06-4

}}

• {{cite book

| last = Rossi | first = Corinna | author-link = Corinna Rossi

| title = Architecture and Mathematics in Ancient Egypt

| year = 2004

| publisher = Cambridge University Press

| pages = 67–68

| url = https://archive.org/details/architechture-and-mathematics-in-ancient-egypt-corianna-rossi-2003/page/67/

}}

• {{cite journal

| last1 = Rossi | first1 = Corinna

| last2 = Tout | first2 = Christopher A.

| year = 2002

| title = Were the Fibonacci series and the Golden Section known in ancient Egypt?

| journal = Historia Mathematica

| volume = 29

| issue = 2

| pages = 101–113

| doi = 10.1006/hmat.2001.2334

| hdl = 11311/997099

| hdl-access = free

}}

• {{cite book

| last = Simonson | first = Shai

| title = Rediscovering Mathematics: You Do the Math

| year = 2011

| url = https://books.google.com/books?id=Ws6-DwAAQBAJ

| publisher = Mathematical Association of America

| isbn = 978-0-88385-912-4

}}

• {{cite journal

| last1 = Slobodan | first1 = Mišić

| last2 = Obradović | first2 = Marija

| last3 = Ðukanović | first3 = Gordana

| title = Composite Concave Cupolae as Geometric and Architectural Forms

| year = 2015

| journal = Journal for Geometry and Graphics

| volume = 19

| issue = 1

| pages = 79–91

| url = https://www.heldermann-verlag.de/jgg/jgg19/j19h1misi.pdf

}}

• {{cite book

| last = Smith | first = James T.

| title = Methods of Geometry

| year = 2000

| url = https://books.google.com/books?id=B0khWEZmOlwC

| publisher = John Wiley & Sons

| isbn = 0-471-25183-6

}}

• {{cite book

| last1 = Takacs | first1 = Sarolta Anna

| last2 = Cline | first2 = Eric H.

| year = 2015

| title = The Ancient World

| publisher = Routledge

| isbn = 978-1-317-45839-5

| page = 16

| url = https://books.google.com/books?id=SPcvCgAAQBAJ&pg=PA16

}}

• {{cite book

| last = Uehara | first = Ryuhei

| year = 2020

| title = Introduction to Computational Origami: The World of New Computational Geometry

| url = https://books.google.com/books?id=51juDwAAQBAJ

| publisher = Springer

| isbn = 978-981-15-4470-5

| doi = 10.1007/978-981-15-4470-5

| s2cid = 220150682

}}

• {{cite journal

| last = Wagner | first = Donald Blackmore

| year = 1979

| title = An early Chinese derivation of the volume of a pyramid: Liu Hui, third century A.D.

| journal = Historia Mathematica

| volume = 6

| issue = 2

| pages = 164–188

| doi = 10.1016/0315-0860(79)90076-4

}}

• {{cite conference

| last = Wohlleben | first = Eva

| editor-last = Cocchiarella | editor-first = Luigi

| year = 2019

| contribution = Duality in Non-Polyhedral Bodies Part I: Polyliner

| conference = International Conference on Geometry and Graphics

| title = ICGG 2018 – Proceedings of the 18th International Conference on Geometry and Graphics: 40th Anniversary – Milan, Italy, August 3–7, 2018

| url = https://books.google.com/books?id=rEpjDwAAQBAJ

| publisher = Springer

| isbn = 978-3-319-95588-9

| doi = 10.1007/978-3-319-95588-9

}}

{{refend}}

{{Commons category}}

• {{MathWorld2|urlname=SquarePyramid|title=Square pyramid|urlname2=JohnsonSolid|title2=Johnson solid}}
• [https://web.archive.org/web/20071008222854/http://polyhedra.org/poly/show/45/square_pyramid Square Pyramid] – Interactive Polyhedron Model
• [https://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] georgehart.com: The Encyclopedia of Polyhedra (VRML [https://www.georgehart.com/virtual-polyhedra/vrml/square_pyramid_(J1).wrl model] {{Webarchive|url=https://web.archive.org/web/20231007033617/https://www.georgehart.com/virtual-polyhedra/vrml/square_pyramid_(J1).wrl |date=7 October 2023 }})

{{Johnson solids}}

Category:Elementary polyhedron

Category:Johnson solids

Category:Prismatoid polyhedra

Category:Pyramids (geometry)

Category:Self-dual polyhedra