Polyhedrons. Prism, parallelepiped, pyramid

Polyhedron is a body, boundary of which consists of pieces of planes (polygons). These polygons are called faces, their sides – edges, their vertices – vertices of  polyhedron. Segments, joining two vertices, which are not placed on the same face, are called diagonals of polyhedron. A polyhedron is called a convex one, if all its diagonals are placed inside of it.

Prism is a polyhedron ( Fig.79 ), two faces of which ABCDE and abcde (bases of prism) are equal polygons with correspondingly parallel sides, and the rest of the faces (AabB, BbcC etc.) are parallelograms, planes of which are parallel  to a straight line (Aa, or Bb, or Cc etc.). Parallelograms AabB, BbcC etc. are called lateral faces;  edges Aa, Bb, Cc etc. are called lateral edges. A height of prism is any perpendicular, drawn from any point of one base to a plane  of another base. Depending on a form of polygon in a base, the prism can be correspondingly: triangular, quadrangular, pentagonal, hexagonal and so on. If lateral edges of a prism are perpendicular to a base plane, this prism is a right prism; otherwise it is an oblique prism. If a base of a right prism is a regular polygon, this prism is also called a regular one. On Fig.79 an oblique pentagonal prism is shown.

Parallelepiped  is a prism, bases of which are parallelograms. So, a parallelepiped has six faces and all of them are parallelograms. Opposite faces are two by  two equal and parallel. A parallelepiped has four diagonals; they all intersect in the one point and they are divided in it into two. If four lateral faces of parallelepiped are rectangles, it is called a right parallelepiped. A right parallelepiped, all six faces of which are rectangles, is called a right-angled parallelepiped. A diagonal of right-angled parallelepiped  d  and its edges  a, b, c  are tied by the relation:  d ² = a ²+ b ² + c ² . A right-angled parallelepiped, all faces of which are squares, is called a cube. All edges of a cube are equal.

Pyramid  is a polyhedron, one face of which (a base of pyramid) is an arbitrary polygon ( ABCDE,  Fig.80 ), and all the rest of the faces (lateral faces) are triangles with a common vertex  S, called a vertex of a pyramid. The perpendicular SO, drawn from a vertex of a pyramid to its base, is called a height of pyramid. Depending on a form of polygon in a base, the pyramid can be correspondingly: triangular, quadrangular, pentagonal, hexagonal and so on. A triangular pyramid is a tetrahedron, a quadrangular one – a pentahedron etc. A pyramid is called a regular one, if its base is a regular polygon and its height falls into a center of a base. All lateral edges of a regular pyramid are equal; all lateral faces are equal isosceles triangles. A height of lateral face ( SF ) is called an apothem of a regular pyramid.

If to draw the section abcde, parallel to the base ABCDE ( Fig.81 ) of the pyramid, then a body, concluded between these planes and lateral surface, is called a truncated pyramid. Parallel faces ABCDE  and  abcde are called its bases; a distance Oo between them is a height. A truncated pyramid is called a regular one, if a pyramid, from which it was received, is regular. All lateral faces of a regular truncated pyramid are equal isosceles trapezoids. The height  Ff  of a lateral face ( Fig.81 ) is called an apothem of a regular truncated pyramid.