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Elongated Pentagonal Pyramid
In geometry, the elongated pentagonal pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal pyramid () by attaching a pentagonal prism to its base. Formulae The following formulae for the height (H), surface area (A) and volume (V) can be used if all faces are regular, with edge length L: :H = L\cdot \left( 1 + \sqrt\right) \approx L\cdot 1.525731112 :A = L^2 \cdot \frac \approx L^2\cdot 8.88554091 :V = L^3 \cdot \left( \frac \right) \approx L^3\cdot 2.021980233 Dual polyhedron The dual of the elongated pentagonal pyramid has 11 faces: 5 triangular, 1 pentagonal and 5 trapezoidal. It is topologically identical to the Johnson solid. See also * Elongated pentagonal bipyramid In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal bipyramid () by inserting a pentagonal prism betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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), vertex. An example of a Johnson solid is the square-based Pyramid (geometry), pyramid with equilateral sides (square pyramid, ); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform polyhedron, uniform (i.e., not Platonic solid, Archimedean solid, prism (geometry), uniform prism, or uniform antiprism) before they refer to it as a “Johnson solid”. As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid () is an example that has a degree-5 vertex. Although there is no obvious restriction tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Prism
In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron If faces are all regular, the pentagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a '' truncated pentagonal hosohedron'', represented by Schläfli symbol t. Alternately it can be seen as the Cartesian product of a regular pentagon and a line segment, and represented by the product ×. The dual of a pentagonal prism is a pentagonal bipyramid. The symmetry group of a right pentagonal prism is ''D5h'' of order 20. The rotation group is ''D5'' of order 10. Volume The volume, as for all prisms, is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base. For a uniform pentagonal pris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Solids
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides ( ); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform (i.e., not Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they refer to it as a “Johnson solid”. As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid () is an example that has a degree-5 vertex. Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johns ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Pentagonal Bipyramid
In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal bipyramid () by inserting a pentagonal prism between its congruent halves. Dual polyhedron The dual of the elongated square bipyramid is a pentagonal bifrustum The pentagonal bifrustum or truncated pentagonal bipyramid is the third in an infinite series of bifrustum polyhedra. It has 10 trapezoid and 2 pentagonal faces. Constructions The pentagonal bifrustum is the dual polyhedron of a Johnson solid, .... See also * Elongated pentagonal pyramid External links * Johnson solids Pyramids and bipyramids {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Elongated Pentagonal Pyramid Net
Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical number), a grammatical category used in some languages * Dual county, a Gaelic games county which in both Gaelic football and hurling * Dual diagnosis, a psychiatric diagnosis of co-occurrence of substance abuse and a mental problem * Dual fertilization, simultaneous application of a P-type and N-type fertilizer * Dual impedance, electrical circuits that are the dual of each other * Dual SIM cellphone supporting use of two SIMs * Aerochute International Dual a two-seat Australian powered parachute design Acronyms and other uses * Dual (brand), a manufacturer of Hifi equipment * DUAL (cognitive architecture), an artificial intelligence design model * DUAL algorithm, or diffusing update algorithm, used to update Internet protocol routing ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Elongated Pentagonal Pyramid
Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical number), a grammatical category used in some languages * Dual county, a Gaelic games county which in both Gaelic football and hurling * Dual diagnosis, a psychiatric diagnosis of co-occurrence of substance abuse and a mental problem * Dual fertilization, simultaneous application of a P-type and N-type fertilizer * Dual impedance, electrical circuits that are the dual of each other * Dual SIM cellphone supporting use of two SIMs * Aerochute International Dual a two-seat Australian powered parachute design Acronyms and other uses * Dual (brand), a manufacturer of Hifi equipment * DUAL (cognitive architecture), an artificial intelligence design model * DUAL algorithm, or diffusing update algorithm, used to update Internet protocol routing ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 definition of length (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. In ancient times, volume is measured using similar-shaped natural containers and later on, standardized containers. Some simple three-dimensional shapes can have its volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. Zero-, one- and two-dimensional objects have no volume; in fourth and higher dimensions, an analogous concept to the normal vo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is about 10,000 m". For example, "Christopher Columbus is 5 foot 2 inches in vertical height." When the term is used to describe vertical position (of, e.g., an airplane) from sea level, height is more often called ''altitude''. Furthermore, if the point is attached to the Earth (e.g., a mountain peak), then altitude (height above sea level) is called ''elevation''. In a two-dimensional Cartesian space, height is measured along the vertical axis (''y'') between a specific point and another that does not have the same ''y''-value. If both points happen to have the same ''y''-value, then their relative height is zero. In the case of three-dimensional space, height is measured along the vertical ''z'' axis, describing a distance from (or "above") t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Pyramid
In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the apex). Like any pyramid, it is self- dual. The ''regular'' pentagonal pyramid has a base that is a regular pentagon and lateral faces that are equilateral triangles. It is one of the Johnson solids (). It can be seen as the "lid" of an icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, More generally an order-2 vertex-uniform pentagonal pyramid can be defined with a regular pentagonal base and 5 isosceles triangle sides of any height. Cartesian coordinates The pentagonal pyramid can be seen as the "lid" of a regular icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, ''J''11. From the Cartesian coordinates 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), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Square Pyramid
In geometry, the elongated square pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a square pyramid () by attaching a cube to its square base. Like any elongated pyramid, it is topologically (but not geometrically) self- dual. Formulae The following formulae for the height (H), surface area (A) and volume (V) can be used if all faces are regular, with edge length L: :H = L\cdot \left( 1 + \frac\right) \approx L\cdot 1.707106781 :A = L^2 \cdot \left( 5 + \sqrt \right) \approx L^2\cdot 6.732050808 :V = L^3 \left( 1 + \frac\right)\approx L^3\cdot 1.23570226 Dual polyhedron The dual of the elongated square pyramid has 9 faces: 4 triangular, 1 square and 4 trapezoidal. Related polyhedra and honeycombs The elongated square pyramid can form a tessellation of space with tetrahedra, similar to a modified tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
