has_facet_polytope dodecahedron
A regular icosahedron has 60 rotational ( or orientation - preserving ) symmetries , and a symmetry order of 120 including transformations that combine a reflection and a rotation . A ___MASK75___ has the same set of symmetries , since it is the dual of the icosahedron . In geometry , a ___MASK60___ ( from the Greek  " pente " and  " gonia ", meaning " five " and " angle ") is any five - sided ___MASK91___ or 5 - gon . The sum of the internal angles in a simple ___MASK60___ is 540 °. In geometry , a ___MASK66___ diagonal of a polyhedron is a diagonal on one of the faces , in contrast to a space diagonal passing through the interior of the polyhedron . In elementary geometry , a polyhedron ( plural polyhedra or polyhedrons ) is a solid in three dimensions with flat polygonal faces , straight edges and sharp corners or vertices . The word polyhedron comes from the Classical Greek  , as " poly -" ( stem of  , " many ") + "- hedron " ( form of  , " base " or " seat "). In three - dimensional space , a Platonic solid is a regular , convex polyhedron . It is constructed by congruent regular polygonal faces with the same number of faces meeting at each ___MASK17___ . Five solids meet those criteria : The Multinational Aircrew Electronic Warfare Tactics Facility Polygone - ___MASK91___ is an Electronic Warfare Tactics Range located on the border between Germany and France . It is a tri - national facility operated by France , Germany and the United States . In geometry a space diagonal ( also interior diagonal , body diagonal or triagonal ) of a polyhedron is a line connecting two vertices that are not on the same ___MASK66___ . Space diagonals contrast with ___MASK66___ diagonals , which connect vertices on the same ___MASK66___ ( but not on the same edge ) as each other . In geometry , a dodecahedron ( Greek  , from  ddeka `` twelve '' +  hédra `` base '' , `` seat '' or `` ___MASK66___ '' ) is any polyhedron with twelve flat faces . The most familiar dodecahedron is the ___MASK75___ , which is a Platonic solid . There are also three regular star dodecahedra , which are constructed as stellations of the convex form . All of these have icosahedral symmetry , order 120 . The pyritohedron is an irregular pentagonal dodecahedron , having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry . The rhombic dodecahedron , seen as a limiting case of the pyritohedron has octahedral symmetry . The elongated dodecahedron and trapezo - rhombic dodecahedron variations , along with the rhombic dodecahedra are space - filling . There are a large number of other dodecahedra . In Euclidean geometry , a regular ___MASK91___ is a ___MASK91___ that is equiangular ( all angles are equal in measure ) and equilateral ( all sides have the same length ). Regular polygons may be convex or star . In the limit , a sequence of regular polygons with an increasing number of sides becomes a circle , if the perimeter is fixed , or a regular apeirogon , if the edge length is fixed . A ___MASK75___ or pentagonal dodecahedron is a dodecahedron that is regular composed of twelve regular pentagonal faces , with three meeting at each vertex , and is represented by the Schläfli symbol { 5 , 3 }. It is one of the five Platonic solids . It has 12 faces , 20 vertices , 30 edges , and 160 diagonals ( 60 face diagonals , 100 space diagonals ).