geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a pentagonal pyramid is a

pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...

with a

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

al base upon which are erected five

triangular A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- collinea ...

faces that meet at a point (the

apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *Apex ...

). Like any

, it is self- dual. The ''regular'' pentagonal pyramid has a base that is a regular pentagon and lateral faces that are

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

s. It is one of the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

s (). It can be seen as the "lid" of an

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

; the rest of the icosahedron forms a

gyroelongated pentagonal pyramid In geometry, the gyroelongated pentagonal pyramid is one of the Johnson solids (). As its name suggests, it is formed by taking a pentagonal pyramid and "gyroelongating" it, which in this case involves joining a pentagonal antiprism to its base ...

, More generally an order-2

vertex-uniform In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

pentagonal pyramid can be defined with a regular pentagonal base and 5

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

sides of any height.

Cartesian coordinates

The pentagonal pyramid can be seen as the "lid" of a

regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It ...

; the rest of the icosahedron forms a

, ''J''₁₁. From the

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

of the icosahedron, Cartesian coordinates for a pentagonal pyramid with edge length 2 may be inferred as :

(1,0,\tau),\,(-1,0,\tau),\,(0,\tau,1),\,(\tau,1,0),(\tau,-1,0),(0,-\tau,1)

where (sometimes written as ''φ'') is the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

. The height ''H'', from the midpoint of the pentagonal face to the apex, of a pentagonal pyramid with edge length ''a'' may therefore be computed as: :

H = \left(\sqrt\right)a \approx 0.52573a.

Its

surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...

''A'' can be computed as the area of the pentagonal base plus five times the area of one triangle: :

A = \frac\sqrt \approx 3.88554 \cdot a^2.

Its

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

can be calculated as: :

V = \left(\frac\right)a^3 \approx 0.30150a^3.

Related polyhedra

The pentagrammic star pyramid has the same

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equ ...

, but connected onto a

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aroun ...

base: :

Example

References

External links

*
Virtual Reality Polyhedra
www.georgehart.com: The Encyclopedia of Polyhedra (

VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graphi ...

br>model
{{Johnson solids navigator Pyramids and bipyramids Self-dual polyhedra Prismatoid polyhedra Johnson solids