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Pyramid (geometry)

 

 
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Pyramid (geometry)
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a pyramid is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 formed by connecting a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
al base and a point, called the apex
Apex (geometry)

In geometry, an apex is a descriptive label for a visual singular highest or most distant point or Vertex in an isosceles triangle, Pyramid or Cone , usually contrasting with the opposite side called the base....
. Each base edge and apex form a triangle. It is a conic solid with polygonal base. Pyramids can have from three to a virtually unlimited amount of sides.

An n-sided pyramid will have n+1 vertices, n+1 faces, and 2n edges. All pyramids are self-dual.

When unspecified, the base is usually assumed to be square.

If the base is a regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
 and the apex is above the center the polygon, an n-gonal pyramid will has Cnv symmetry.

Pyramids are a subclass of the prismatoid
Prismatoid

A prismatoid is a polyhedron where all vertices lie in two parallel planes. If the areas of the two parallel faces are A1 and A3, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A2, and the height is h, then the volume of the prismatoid i...
s.

Pyramids with regular polygon faces
The regular
Regular polyhedron

A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
 tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
, one of the Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s, is a triangular pyramid all of whose faces are equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a pyramid is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 formed by connecting a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
al base and a point, called the apex
Apex (geometry)

In geometry, an apex is a descriptive label for a visual singular highest or most distant point or Vertex in an isosceles triangle, Pyramid or Cone , usually contrasting with the opposite side called the base....
. Each base edge and apex form a triangle. It is a conic solid with polygonal base. Pyramids can have from three to a virtually unlimited amount of sides.

An n-sided pyramid will have n+1 vertices, n+1 faces, and 2n edges. All pyramids are self-dual.

When unspecified, the base is usually assumed to be square.

If the base is a regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
 and the apex is above the center the polygon, an n-gonal pyramid will has Cnv symmetry.

Pyramids are a subclass of the prismatoid
Prismatoid

A prismatoid is a polyhedron where all vertices lie in two parallel planes. If the areas of the two parallel faces are A1 and A3, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A2, and the height is h, then the volume of the prismatoid i...
s.

Pyramids with regular polygon faces


The regular
Regular polyhedron

A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
 tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
, one of the Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s, is a triangular pyramid all of whose faces are equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s. Besides the triangular pyramid, only the square and pentagonal pyramids can be composed of equilateral triangles, and in that case they are Johnson solid
Johnson solid

In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
s.

Tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
Square pyramid
Square pyramid

In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
Pentagonal pyramid
Pentagonal pyramid

In geometry, a pentagonal pyramid is a Pyramid with a pentagonal base upon which are erected five triangle faces that meet at a point . Like any pyramid, it is self-dual polyhedron....
Square Pyramid
Pentagonal Pyramid


Star pyramids


Pyramids with regular star polygon bases can also be constructed.

For example the pentagrammic pyramid has a pentagram
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
 base and 5 intersecting equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
 sides.

Volume


The volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
 of a pyramid is where B is the area of the base and h the height from the base to the apex. This works for any location of the apex, provided that h is measured as the perpendicular
Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
 distance from the plane which contains the base.

One way to find the Volume of any Pyramid is to find the Volume of any Prism and then divide it by 3

This can be proven using calculus:
It can be proved using similarity that the dimensions of a cross section parallel to the base increase linearly from the apex to the base. Then, the cross section at any height y is the base scaled by a factor of , where h is the height from the base to the apex. Since the area of any shape is multiplied by the square of the shape's scaling factor, the area of a cross section at height y is .
The volume is given by the integral


(Trivially, the volume of a square-based pyramid with an apex half the height of its base can be seen to correspond to 1/6 of a cube formed by fitting 6 such pyramids (in opposite pairs) about a center. Since the "base times height" then corresponds to one half of the cube's volume it is therefore 3 times the volume of the pyramid and the factor of 1/3 follows.)

Surface area


The surface area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 of a regular pyramid is where is the area of the base, p is the perimeter of the base, and s is the slant height along the bisector of a face (ie the length from the midpoint of any edge of the base to the apex).

  • Bipyramid
    Bipyramid

    An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal Pyramid and its mirror image base-to-base.The referenced n-agon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the 2 pyramid halves....
  • Cone (geometry)
    Cone (geometry)

    A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
  • Trigonal pyramid (chemistry)
    Trigonal pyramid (chemistry)

    In chemistry, a trigonal pyramid is a molecular geometry with one atom at the apex and three atoms at the corners of a trigonal base. When all three atoms at the corners are identical, the molecule belongs to Molecular symmetry C3v....


External links

  • at
  • The Encyclopedia of Polyhedra
    • VRML
      VRML

      VRML is a standard file format for representing 3-D computer graphics interactive vector graphics, designed particularly with the World Wide Web in mind....
       models