Square pyramid

Square pyramid
In geometry, a square pyramid is a pyramid with a square base, having a total of five faces. If the pyramid's apex lies on a line erected perpendicularly from 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 arisen throughout the history of architecture, with an example being the Egyptian pyramid. They also crop up in chemistry in square pyramidal molecular structures. Square pyramids are often used in the construction of other polyhedra.
Properties
Right and equilateral square pyramid
A square pyramid has eight edges, five faces that include four triangles, and one square as the base. It also has five vertices, one of which is an apex, a vertex where all lateral edges meet. A lateral edge is a segment line between the apex and another vertex at the square base. When the apex is perpendicularly above the center of the base, all the lateral edges have the same length, and the faces other than the base are congruent isosceles triangles, a right square pyramid. A square pyramid where the apex is not perpendicularly above the center of the base, and the faces are not an isosceles triangle is an oblique square pyramid.
When all edges have the same length, the four isosceles triangles become equilateral, and therefore all the faces are regular polygons. This pyramid is known as an equilateral square pyramid. Its dihedral angle between two adjacent triangular faces and between the base and triangular face is
respectively. It has three-dimensional symmetry group of cyclic group pyramidal symmetry, a symmetry of order eight; this means that it is symmetrical as one rotates it for every quarter-turn of a full angle around the axis of symmetry, two vertical planes passing through the diagonals of the square base, and two other vertical planes passing through the midpoint of the opposite edges of the square base. A convex polyhedra that has regular polygons as its faces is called Johnson solid, and because an equilateral square pyramid has these properties, it is a Johnson solid that numbered as among them.
Surface area and volume
The slant height of a right square pyramid can be considered as the height of an isosceles triangle, which can be obtained by applying the Pythagorean theorem, that is:
Here, is the length of the base of an isosceles triangle, and is the length of the side of an isosceles triangle, a lateral edge of a right square pyramid. The height of a right square pyramid can be considered and obtained similarly, and substitute the formula of the slant height gives
A surface area is the total area of the faces of a polyhedron. Hence, the surface area of a square pyramid can be expressed as the sum of four times the area of a triangle and the area of a square. The area of a triangle is the half product of the base and side, and the area of a square is the second power of its side. This can be formulated as:
The volume of a pyramid, in general, is given by one-third of the product of the area of the base and the height. As for the pyramid with a square base, the formula can be expressed as:
Applications
In architecture, an example of a building that resembles a square pyramid is Egyptian pyramid. In Moscow Mathematical Papyrus, the ancient Egyptians already knew the formula for calculating the volume of a truncated square pyramid. One can conclude that they were already acquainted with the volume of a square pyramid, but it is unknown how the formula was attested. The problem of finding the slope and height of a square pyramid can be found in the Rhind Mathematical Papyrus. Other buildings that has the structure of a square pyramid are the Louvre Pyramid and the casino hotel Luxor Las Vegas.
In stereochemistry, a square pyramid can be described as the cluster of a square pyramidal molecular geometry. The square pyramidal molecule is the main-group element that has one active lone pair, which can be described by VSEPR theory, a model to predict the arrangement of the number of electron pairs around central atoms in molecules. Some examples of a cluster based on that structure are chlorine pentafluoride, bromine pentafluoride, and iodine pentafluoride.
A square pyramid can be used to construct a new polyhedron by attaching its base onto the square face of another polyhedron, a process known as augmentation. In general, it is a process of attaching the pyramid, cupolae, and rotunda to another polyhedron; specifically, a process of attaching them to a prism and an antiprism are known as elongation and gyroelongation, respectively. An example of this construction is an octahedron (or a square bipyramid), which can be constructed by attaching the base of two square pyramids. Another polyhedron is a tetrakis hexahedron, which can be constructed by putting the base of six equilateral square pyramids onto the faces of a cube. In addition to those polyhedrons, some Johnson solids can be constructed by attaching the base of some square pyramids as well, that is: 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 .
See also
Square pyramidal number, a natural number that counts the number of stacked spheres in a square pyramid.
References
External links
Square Pyramid – Interactive Polyhedron Model
Virtual Reality Polyhedra georgehart.com: The Encyclopedia of Polyhedra (VRML model)
Johnson solids
Prismatoid polyhedra
Pyramids and bipyramids
Self-dual polyhedra