Square pyramid | |
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Type | Pyramid, Johnson J92 – J1 – J2 |
Faces | 4 triangles 1 square |
Edges | 8 |
Vertices | 5 |
Vertex configuration | [1] |
Symmetry group | |
Volume | |
Dihedral angle (degrees) | Equilateral square pyramid: [1]
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Dual polyhedron | self-dual |
Properties | convex, elementary (equilateral square pyramid) |
Net | |
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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. It is called an equilateral square pyramid, an example of a Johnson solid.
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.
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. [2] 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. [3] 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. [4]
The slant height of a right square pyramid is defined as the height of one of its isosceles triangles. It can be obtained via the Pythagorean theorem: where is the length of the triangle's base, also one of the square's edges, and is the length of the triangle's legs, which are lateral edges of the pyramid. [5] The height of a right square pyramid can be similarly obtained, with a substitution of the slant height formula giving: [6] A polyhedron's surface area is the sum of the areas of its faces. The surface area of a right square pyramid can be expressed as , where and 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: [7] In general, the volume of a pyramid is equal to one-third of the area of its base multiplied by its height. [8] Expressed in a formula for a square pyramid, this is: [9]
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. [10] The Babylonian mathematicians also considered the volume of a frustum, but gave an incorrect formula for it. [11] One Chinese mathematician Liu Hui also discovered the volume by the method of dissecting a rectangular solid into pieces. [12]
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. [13] The dihedral angles between adjacent triangular faces are , and that between the base and each triangular face being half of that, . [1] 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 . [14]
Because its edges are all equal in length (that is, ), its slant, height, surface area, and volume can be derived by substituting the formulas of a right square pyramid: [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 : 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. [1] It can be represented as the wheel graph ; more generally, a wheel graph is the representation of the skeleton of a -sided pyramid. [16] It is self-dual, meaning its dual polyhedron is the square pyramid itself. [17]
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. [18]
In architecture, the pyramids built in ancient Egypt are examples of buildings shaped like square pyramids. [19] 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. [20] The Mesoamerican pyramids are also ancient pyramidal buildings similar to the Egyptian; they differ in having flat tops and stairs ascending their faces. [21] Modern buildings whose designs imitate the Egyptian pyramids include the Louvre Pyramid and the casino hotel Luxor Las Vegas. [22]
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. [23] Examples of molecules with this structure include chlorine pentafluoride, bromine pentafluoride, and iodine pentafluoride. [24]
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. [25] Attaching prisms or antiprisms to pyramids is known as elongation or gyroelongation, respectively. [26] Some of the other Johnson solids can be constructed by either augmenting square pyramids or augmenting other shapes with square pyramids: elongated square pyramid , gyroelongated square pyramid , elongated square bipyramid , gyroelongated square bipyramid , augmented triangular prism , biaugmented triangular prism , triaugmented triangular prism , augmented pentagonal prism , biaugmented pentagonal prism , augmented hexagonal prism , parabiaugmented hexagonal prism , metabiaugmented hexagonal prism , triaugmented hexagonal prism , and augmented sphenocorona . [27]
In geometry, an octahedron is a polyhedron with eight faces. An octahedron can be considered as a square bipyramid. When the edges of a square bipyramid are all equal in length, it produces a regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. It is also an example of a deltahedron. An octahedron is the three-dimensional case of the more general concept of a cross polytope.
In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid.
The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.
In geometry, the pentagonal bipyramid is a polyhedron with 10 triangular faces. It is constructed by attaching two pentagonal pyramids to each of their bases. If the triangular faces are equilateral, the pentagonal bipyramid is an example of deltahedra, and of Johnson solid.
In geometry, pentagonal pyramid is a pyramid with a pentagon base and five triangular faces, having a total of six faces. It is categorized as Johnson solid if all of the edges are equal in length, forming equilateral triangular faces and a regular pentagonal base. The pentagonal pyramid can be found in many polyhedrons, including their construction. It also occurs in stereochemistry in pentagonal pyramidal molecular geometry.
In geometry, the triangular cupola is the cupola with hexagon as its base and triangle as its top. If the edges are equal in length, the triangular cupola is the Johnson solid. It can be seen as half a cuboctahedron. The triangular cupola can be applied to construct many polyhedrons.
In geometry, the elongated square cupola is a polyhedron constructed from an octagonal prism by attaching square cupola onto its base. It is an example of Johnson solid.
In geometry, the elongated square gyrobicupola is a polyhedron constructed by two square cupolas attaching onto the bases of octagonal prism, with one of them rotated. It was once mistakenly considered a rhombicuboctahedron by many mathematicians. It is not considered to be an Archimedean solid because it lacks a set of global symmetries that map every vertex to every other vertex, unlike the 13 Archimedean solids. It is also a canonical polyhedron. For this reason, it is also known as pseudo-rhombicuboctahedron, Miller solids, or Miller–Askinuze solid.
In geometry, the elongated triangular pyramid is one of the Johnson solids. As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically self-dual.
In geometry, the elongated triangular bipyramid or triakis triangular prism a polyhedron constructed from a triangular prism by attaching two tetrahedrons to its bases. It is an example of Johnson solid.
In geometry, the elongated pentagonal bipyramid is a polyhedron constructed by attaching two pentagonal pyramids onto the base of a pentagonal prism. It is an example of Johnson solid.
In geometry, the gyrobifastigium is a polyhedron that is constructed by attaching a triangular prism to square face of another one. It is an example of a Johnson solid. It is the only Johnson solid that can tile three-dimensional space.
In geometry, the augmented triangular prism is a polyhedron constructed by attaching an equilateral square pyramid onto the square face of a triangular prism. As a result, it is an example of Johnson solid. It can be visualized as the chemical compound, known as capped trigonal prismatic molecular geometry.
In geometry, the biaugmented triangular prism is a polyhedron constructed from a triangular prism by attaching two equilateral square pyramids onto two of its square faces. It is an example of Johnson solid. It can be found in stereochemistry in bicapped trigonal prismatic molecular geometry.
In geometry, the augmented pentagonal prism is a polyhedron that can be constructed by attaching an equilateral square pyramid onto the square face of pentagonal prism. It is an example of Johnson solid.
In geometry, the biaugmented pentagonal prism is a polyhedron constructed from a pentagonal prism by attaching two equilateral square pyramids onto each of its square faces. It is an example of Johnson solid.
In geometry, the augmented hexagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid to one of its equatorial faces. When two or three such pyramids are attached, the result may be a parabiaugmented hexagonal prism, a metabiaugmented hexagonal prism, or a triaugmented hexagonal prism.
In geometry, the elongated triangular cupola is a polyhedron constructed from a hexagonal prism by attaching a triangular cupola. It is an example of a Johnson solid.
In geometry, the elongated triangular orthobicupola is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid.
In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.