Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with a polygonal base. Many types of pyramids can be found by determining the shape of bases, or cutting off the apex. It can be generalized into higher dimension, known as hyperpyramid. All pyramids are self-dual.

Etymology
The word "pyramid" derives from the ancient Greek term "πυραμίς" (pyramis), which referred to a pyramid-shaped structure and a type of wheat cake. The term is rooted in the Greek "πυρ" (pyr, 'fire') and "άμις" (amis, 'vessel'), highlighting the shape's pointed, flame-like appearance.

In Byzantine Greek, the term evolved to "πυραμίδα" (pyramída), continuing to denote pyramid structures. The Greek term "πυραμίς" was borrowed into Latin as "pyramis." The term "πυραμίδα" influenced the evolution of the word into "pyramid" in English and other languages.

Definition
A pyramid is a polyhedron that may be formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form an isosceles triangle, called a lateral face. The edges connected from the polygonal base's vertices to the apex are called lateral edges. Historically, the definition of a pyramid has been described by many mathematicians in ancient times. Euclides in his Elements defined a pyramid as a solid figure, constructed from one plane to one point. The context of his definition was vague until Heron of Alexandria defined it as the figure by putting the point together with a polygonal base.

A prismatoid is defined as a polyhedron where its vertices lie on two parallel planes, with its lateral faces are triangles, trapezoids, and parallelograms. Pyramids are classified as prismatoid.

Classification and types
A right pyramid is a pyramid where the base is circumscribed about the circle and the altitude of the pyramid meets at the circle's center. This pyramid may be classified based on the regularity of its bases. A pyramid with a regular polygon as the base is called a regular pyramid. For the pyramid with an $n$-sided regular base, it has $n + 1$ vertices, $n + 1$ faces, and $2n$ edges. Such pyramid has isosceles triangles as its faces, with its symmetry is $C_{nv}$, a symmetry of order $2n$: the pyramids are symmetrical as they rotated around their axis of symmetry (a line passing through the apex and the base centroid), and they are mirror symmetric relative to any perpendicular plane passing through a bisector of the base. Examples are square pyramid and pentagonal pyramid, a four- and five-triangular faces pyramid with a square and pentagon base, respectively; they are classified as the first and second Johnson solid if their regular faces and edges that are equal in length, and their symmetries are $C_{4v}$ of order 8 and $C_{5v}$ of order 10, respectively. A tetrahedron or triangular pyramid is an example that has four equilateral triangles, with all edges equal in length, and one of them is considered as the base. Because the faces are regular, it is an example of a Platonic solid and deltahedra, and it has tetrahedral symmetry. A pyramid with the base as circle is known as cone. Pyramids have the property of self-dual, meaning their duals are the same as vertices corresponding to the edges and vice versa. Their skeleton may be represented as the wheel graph.

A right pyramid may also have a base with an irregular polygon. Examples are the pyramids with rectangle and rhombus as their bases. These two pyramids have the symmetry of $C_{2v}$ of order 4.

The type of pyramids can be derived in many ways. The base regularity of a pyramid's base may be classified based on the type of polygon, and one example is the pyramid with regular star polygon as its base, known as the star pyramid. The pyramid cut off by a plane is called a truncated pyramid; if the truncation plane is parallel to the base of a pyramid, it is called a frustum.

Mensuration
The surface area is the total area of each polyhedra's faces. In the case of a pyramid, its surface area is the sum of the area of triangles and the area of the polygonal base.

The volume of a pyramid is the one-third product of the base's area and the height. Given that $$ B $$ is the base's area and $$ h $$ is the height of a pyramid. Mathematically, the volume of a pyramid is: $$ V = \frac{1}{3}Bh. $$ The volume of a pyramid was recorded back in ancient Egypt, where they calculated the volume of a square frustum, suggesting they acquainted the volume of a square pyramid. The formula of volume for a general pyramid was discovered by Indian mathematician Aryabhata, where he quoted in his Aryabhatiya that the volume of a pyramid is incorrectly the half product of area's base and the height.

Generalization
The hyperpyramid is the generalization of a pyramid in $n$-dimensional space. In the case of the pyramid, one connects all vertices of the base, a polygon in a plane, to a point outside the plane, which is the peak. The pyramid's height is the distance of the peak from the plane. This construction gets generalized to $n$ dimensions. The base becomes a $(n − 1)$-polytope in a $(n − 1)$-dimensional hyperplane. A point called the apex is located outside the hyperplane and gets connected to all the vertices of the polytope and the distance of the apex from the hyperplane is called height.

The $n$-dimensional volume of a $n$-dimensional hyperpyramid can be computed as follows: $$ V_n = \frac{A  \cdot h}{n}. $$ Here $V_{n}$ denotes the $n$-dimensional volume of the hyperpyramid. $A$ denotes the $(n − 1)$-dimensional volume of the base and $h$ the height, that is the distance between the apex and the $(n − 1)$-dimensional hyperplane containing the base $A$.