In
geometry, a square pyramid is a
pyramid with a square base, having a total of five faces. If the
apex of the pyramid is directly above the center of the square, it is a right square pyramid with four
isosceles triangles; otherwise, it is an oblique square pyramid. When all of the pyramid's edges are equal in length, its triangles are all
equilateral, and it is called an equilateral square pyramid.
Square pyramids have appeared throughout the history of architecture, with examples being
Egyptian pyramids, and many other similar buildings. They also occur in chemistry in
square pyramidal molecular structures. Square pyramids are often used in the construction of other
polyhedra. Many mathematicians in ancient times discovered the formula for the volume of a square pyramid with different approaches.
Properties
Right square pyramid
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.^{
[3]} 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.^{
[4]} 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.^{
[5]}
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:
$s={\sqrt {b^{2}-{\frac {l^{2}}{4}}}},$
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.^{
[6]} The height $h$ of a right square pyramid can be similarly obtained, with a substitution of the slant height formula giving:^{
[7]}
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:^{
[8]}
In general, the volume $V$ of a pyramid is equal to one-third of the area of its base multiplied by its height.^{
[9]} Expressed in a formula for a square pyramid, this is:^{
[10]}
$V={\frac {1}{3}}l^{2}h.$
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.^{
[11]} The Babylonian mathematicians also considered the volume of a frustum, but gave an incorrect formula for it.^{
[12]} One Chinese mathematician
Liu Hui also discovered the volume by the method of dissecting a rectangular solid into pieces.^{
[13]}
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.^{
[14]} The
dihedral angles between adjacent triangular faces are $\arccos \left(-1/3\right)\approx 109.47^{\circ }$, and that between the base and each triangular face being half of that, $\arctan {\sqrt {2}}\approx 54.735^{\circ }$.^{
[1]} A
convex polyhedron with only regular polygons as faces is called a
Johnson solid, and the equilateral square pyramid is the first Johnson solid, enumerated as $J_{1}$.^{
[15]} 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$C_{4v}$: 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. It is also
mirror symmetric relative to any perpendicular plane passing through a bisector of the base.^{
[1]} It can be represented as the
wheel graph$W_{4}$; more generally, a wheel graph $W_{n}$ is the representation of the
skeleton of a $n$-sided pyramid.^{
[16]}
Because all edges of the equilateral square pyramid are 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:^{
[17]}
The
Egyptian pyramids are examples of square pyramidal buildings in architecture.
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.^{
[18]}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.^{
[19]} The
Mesoamerican pyramids are also ancient pyramidal buildings similar to the Egyptian; they differ in having flat tops and stairs ascending their faces.^{
[20]} Modern buildings whose designs imitate the Egyptian pyramids include the
Louvre Pyramid and the casino hotel
Luxor Las Vegas.^{
[21]}
^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
Rossi (2004), pp.
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 $\varphi$, and $\varphi$ 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
Rossi & Tout (2002) and
Markowsky (1992).
Jarvis, Daniel; Naested, Irene (2012). Exploring the Math and Art Connection: Teaching and Learning Between the Lines. Brush Education.
ISBN978-1-55059-398-3.
Wagner, Donald Blackmore (1979). "An early Chinese derivation of the volume of a pyramid: Liu Hui, third century A.D.". Historia Mathematics. 6 (2): 164–188.
doi:
10.1016/0315-0860(79)90076-4.