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This shape is also called Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron. The snub disphenoid can be visualized as an atom cluster surrounding a central atom, that is the dodecahedral molecular geometry .
Its surface area is four times the area of an equilateral triangle: = =. [7] Its volume can be ascertained similarly as the other pyramids, one-third of the base times height. Because the base is an equilateral, it is: [ 7 ] V = 1 3 ⋅ ( 3 4 a 2 ) ⋅ 6 3 a = a 3 6 2 ≈ 0.118 a 3 . {\displaystyle V={\frac {1}{3}}\cdot \left({\frac {\sqrt {3 ...
Alternatively, if you expand each of five cubes by moving the faces away from the origin the right amount and rotating each of the five 72° around so they are equidistant from each other, without changing the orientation or size of the faces, and patch the pentagonal and triangular holes in the result, you get a rhombicosidodecahedron ...
The base regularity of a pyramid's base may be classified based on the type of polygon: one example is the star pyramid in which its base is the regular star polygon. [28] The truncated pyramid is a pyramid cut off by a plane; if the truncation plane is parallel to the base of a pyramid, it is called a frustum.
A dodecagonal pyramid is a pyramid with a dodecagonal base. It is a type of tridecahedron, which has 13 faces , 24 edges , and 13 vertices , and its dual polyhedron is itself. [ 5 ] A regular dodecagonal pyramid is a dodecagonal pyramid whose base is a regular dodecagon.
In geometry, a hyperpyramid is a generalisation of the normal pyramid to n dimensions. 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 generalised to n dimensions.
An elongated triangular pyramid with edge length has a height, by adding the height of a regular tetrahedron and a triangular prism: [4] (+). Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares: [2] (+), and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up: [2]: ((+)).
Pentagonal pyramids are added to the 12 pentagonal faces of the snub dodecahedron, and triangular pyramids are added to the 20 triangular faces that do not share an edge with a pentagon. The pyramid heights are adjusted to make them coplanar with the other 60 triangular faces of the snub dodecahedron. The result is the pentagonal ...