Tuesday, April 28, 2015

Unizor - Geometry3D - Elements - Prisms

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Consider a plane α that we will call a base plane and a polygon ABCDEF on this plane (we specify 6-sided polygon, but it's not important, any polygon will do). This polygon will be used as a directrix of a cylindrical surface we will construct.
Also assume there is another plane β parallel to a plane α that we will also call a base plane. To differentiate between two base planes, we will call one of them "bottom" and another - "top".
Finally, assume that we have a straight line d that is not parallel to base planes.

Let's construct a cylindrical surface σ using polygon ABCDEF as a directrix and line d as a generatrix.

Prism is an object in solid geometry formed by a part of a cylindrical surface σ between base planes α and β and parts of the base planes inside that cylindrical surface - polygon ABCDEF on the "bottom" base plane α and a corresponding polygon A'B'C'D'E'F' on the "top" base plane β.

Points A, B, A', B' and all others are called vertices of a prism.
Segments AB, BC, EE' and all others are called edges of a prism.
Polygons ABCDEF and A'B'C'D'E'F' are, as we mentioned before, called bases of a prism.
Polygons ABB'A', BCC'B' and others are called sides of a prism.
Bases and sides of a prism are generically referred to as faces. Sides are also referred to as lateral faces.

It's easy to prove (and we will do it in one of the future lectures) that all sides ABB'A', BCC'B' etc. are parallelograms and bases ABCDEF and A'B'C'D'E'F' are congruent to each other.

If a generatrix is perpendicular to bases (and, therefore, all side edges, like AA', BB' etc. are also perpendicular to bases) then a prism is called a right prism. All prisms that do not have this characteristic, that is if a generatrix is not perpendicular to bases, are called oblique prisms.

Another classification of prisms is by their bases.
If a base is a triangle, the prism is called triangular prism.
If a base is a rectangle, the prism is called rectangular prism.
If a base is a parallelogram, the prism is called parallelepiped.
If a base is a square and it's a right prism and all edges are of the same length, the prism is called cube.
If a base is a hexagon, the prism is called hexagonal prism.
If a base is a regular N-sided polygon and it's a right prism, it might be called N-prism.

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