Polyhedra Topic Index | Geometry Index | Regents Exam Prep Center

A polyhedron is a three-dimensional solid figure in which each side is a flat surface.  These flat surfaces are polygons and are joined at their edges.   The word polyhedron is derived from the Greek poly (meaning many) and the Indo-European hedron (meaning seat or face).

A polyhedron has no curved surfaces.

The common polyhedron are pyramids and prisms.

pyramid

prism

  A polyhedron is called regular if the faces are congruent, regular polygons and the same number of faces meet at each vertex.  There are a total of five such convex regular polyhedra called the Platonic solids.

tetrahedron

octahedron

icosahedron

hexahedron

dodecahedron

Euler's Polyhedron Theorem:
   Euler discovered that the number of faces (flat surfaces) plus the number of vertices (corner points) of a polyhedron equals the number of edges of the polyhedron plus 2.
                                                          F + V = E + 2

Non-Polyhedra
  The following solids are not polyhedra since a part or all of the figure is curved.

Cylinder

Cone

Sphere

Torus

A torus is a "tube shape".  Examples include an inner tube, a doughnut, a tire and a bagel.  Small r is the radius of the tube and capital R is the distance from the centre of the torus to the center of the tube.

While the torus has a hole in the center, the Surface Area:  the Volume: