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Prismatoid with parallel faces A 1 and A 3, midway cross-section A 2, and height h. In geometry, a prismatoid is a polyhedron whose vertices all lie in two parallel planes.Its lateral faces can be trapezoids or triangles. [1]
If a plane intersects a solid (a 3-dimensional object), then the region common to the plane and the solid is called a cross-section of the solid. [1] A plane containing a cross-section of the solid may be referred to as a cutting plane. The shape of the cross-section of a solid may depend upon the orientation of the cutting plane to the solid.
For example, the icosahedron is {3,5+} 1,0, and pentakis dodecahedron, {3,5+} 1,1 is seen as a regular dodecahedron with pentagonal faces divided into 5 triangles. The primary face of the subdivision is called a principal polyhedral triangle (PPT) or the breakdown structure .
The surface of a convex polyhedron has a metric that is locally Euclidean except at the vertices of the polyhedron, and a curve that avoids the vertices is a geodesic if it follows straight line segments within each face of the polyhedron and stays straight across each polyhedron edge that it crosses.
where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. [2] Any convex polyhedron's surface has Euler characteristic = + = . This equation, stated by Euler in 1758, [3] is known as Euler's polyhedron formula. [4]
A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its interior. Solid geometry deals with the measurements of volumes of various solids, including pyramids , prisms (and other polyhedrons ), cubes , cylinders , cones (and truncated cones ).
For example a tetrahedron is a polyhedron with four faces, a pentahedron is a polyhedron with five faces, a hexahedron is a polyhedron with six faces, etc. [16] For a complete list of the Greek numeral prefixes see Numeral prefix § Table of number prefixes in English, in the column for Greek cardinal numbers.
The convex polyhedron means as bounded intersections of finitely many half-spaces, or as the convex hull of finitely many points. [3] Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors required that Johnson solids are not uniform .