## Tuesday, June 30, 2015

### Unizor - Geometry3D - Prisms - Parallelepiped

Unizor - Creative Minds through Art of Mathematics - Math4Teens

An introduction to prisms was presented in the chapter "Elements of Solid Geometry" of this course (see topics "Cylindrical Surface" and "Prisms"). We recommend to refresh this information prior to continue with properties and characteristics of prisms presented in this lecture.

Parallelepiped is a prism with a parallelogram as its directrix.

Arguably, parallelepiped is the most frequently occurring type of prisms, especially, if it's a rectangular parallelepiped (see below).

Theorem 1
All faces of a parallelepiped are parallelograms.

Side diagonal is a diagonal of a side parallelogram (like AB' or BC').
Base diagonal is a diagonal of a base parallelogram (like AC or B'D').
Space diagonal is a diagonal connecting vertices of opposite bases lying inside a parallelepiped (that is, AC', BD', CA' and DB').

Right parallelepipeds (a particular case of parallelepipeds) are parallelepipeds with all side edges (AA', BB', CC' and DD') perpendicular to bases. All side faces of right parallelepiped are rectangular since side edges are perpendicular to base edges.

Rectangular parallelepipeds or cuboids (a particular case of right parallelepipeds) are right parallelepipeds with rectangular bases. So, all faces of rectangular parallelepiped are rectangular and all edges are perpendicular to sides that are not parallel to them (for instance, AB⊥AA'D'D, B'C'⊥CC'D'D etc.).

Cubes (a particular case of rectangular parallelepipeds) are parallelepipeds with all square faces.

All parallelepipeds fall into category of hexahedrons because they have six faces.

Obviously, all opposite faces of a parallelepiped are congruent and parallel since they are parallelograms with sides correspondingly parallel and equal in length.

Theorem 2
All space diagonals of a parallelepiped AC', BD', CA' and DB' intersect in one point and are divided by that point in half.

The following is a kind of three-dimensional equivalent of Pythagorean Theorem.

Theorem 3
In a rectangular parallelepiped a square of a space diagonal equals to a sum of squares of three edges that share a vertex with this diagonal.

Rectangular parallelepiped is fully defined by the length of any three edges sharing the same vertex (for instance, AB, AD and AA'), because all other edges are equal to one of them and all angles are right angles.

The area of each face of a rectangular parallelepiped is the area of a rectangle, that is a product of the length of two of its sides sharing the same vertex.
If three edges sharing a vertex A are a, b and c, the combined area of all six faces of a rectangular parallelepiped equals to
2ab+2bc+2ac=2(ab+bc+ac)