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Pyramid (geometry)
A pyramid is a polyhedron (a geometric figure) formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. A pyramid is a conic solid with a polygonal base. Many types of pyramids can be found by determining the shape of bases, either by based on a regular polygon (regular pyramids) or by cutting off the apex (truncated pyramid). It can be generalized into higher dimensions, known as hyperpyramid. All pyramids are self-dual.
A pyramid is a polyhedron that may be formed by connecting each vertex in a planar polygon to a point lying outside that plane. This point is called the pyramid's apex, and the planar polygon is the pyramid's base. Each other face of the pyramid is a triangle consisting of one of the base's edges, and the two edges connecting that edge's endpoints to the apex. These faces are called the pyramid's lateral faces, and each edge connected to the apex is called a lateral edge. 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 as triangles, trapezoids, and parallelograms. Pyramids are classified as prismatoid.
The terms "right pyramid" and "regular pyramid" are used to describe special cases of pyramids. Their common notions are as follows. A regular pyramid is one with a regular polygon as its base. A right pyramid is one where the axis (the line joining the centroid of the base and the apex) is perpendicular to the base. An oblique pyramid is one where the axis is not perpendicular to the base. However, there are no standard definitions for these terms, and different sources use them somewhat differently.
Some sources define the term "right pyramid" only as a special case for regular pyramids, while others define it for the general case of any shape of a base. Other sources define only the term "right pyramid" to include within its definition the regular base. Rarely, a "right pyramid" is defined to be a pyramid whose base is circumscribed about a circle and the altitude of the pyramid meets the base at the circle's center.
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 Cnv, 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 C4v of order 8 and C5v 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, that is they can be depicted as a polygon in which its vertices connect a vertex in the center called the universal vertex.
A right pyramid may also have a base with an irregular polygon. Examples of irregular pyramids are those with rectangle and rhombus as their bases. These two pyramids have the symmetry of C2v of order 4.
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Pyramid (geometry)
A pyramid is a polyhedron (a geometric figure) formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. A pyramid is a conic solid with a polygonal base. Many types of pyramids can be found by determining the shape of bases, either by based on a regular polygon (regular pyramids) or by cutting off the apex (truncated pyramid). It can be generalized into higher dimensions, known as hyperpyramid. All pyramids are self-dual.
A pyramid is a polyhedron that may be formed by connecting each vertex in a planar polygon to a point lying outside that plane. This point is called the pyramid's apex, and the planar polygon is the pyramid's base. Each other face of the pyramid is a triangle consisting of one of the base's edges, and the two edges connecting that edge's endpoints to the apex. These faces are called the pyramid's lateral faces, and each edge connected to the apex is called a lateral edge. 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 as triangles, trapezoids, and parallelograms. Pyramids are classified as prismatoid.
The terms "right pyramid" and "regular pyramid" are used to describe special cases of pyramids. Their common notions are as follows. A regular pyramid is one with a regular polygon as its base. A right pyramid is one where the axis (the line joining the centroid of the base and the apex) is perpendicular to the base. An oblique pyramid is one where the axis is not perpendicular to the base. However, there are no standard definitions for these terms, and different sources use them somewhat differently.
Some sources define the term "right pyramid" only as a special case for regular pyramids, while others define it for the general case of any shape of a base. Other sources define only the term "right pyramid" to include within its definition the regular base. Rarely, a "right pyramid" is defined to be a pyramid whose base is circumscribed about a circle and the altitude of the pyramid meets the base at the circle's center.
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 Cnv, 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 C4v of order 8 and C5v 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, that is they can be depicted as a polygon in which its vertices connect a vertex in the center called the universal vertex.
A right pyramid may also have a base with an irregular polygon. Examples of irregular pyramids are those with rectangle and rhombus as their bases. These two pyramids have the symmetry of C2v of order 4.