What is the name of a polyhedron whose bases are rectangles and whose other faces are parallelograms?

A prism is a polyhedron whose faces consist of two congruent polygons lying in parallel planes and a number of parallelograms. The sides of the parallelograms are the segments that join the corresponding vertices of the two congruent polygons. These two congruent polygons are called the bases of the prism. The parallelograms are called the lateral faces of the prism. The segments that join the bases and form the sides of the lateral faces are called the lateral edges of the prism. The union of the two polygons and the parallelograms form the entire prism.

Some obvious questions come up at this point. How many lateral faces are in a prism? The number of lateral faces is equal to the number of sides in the bases. If the bases are quadrilaterals, for example, then there will be four lateral faces. Why are the lateral faces parallelograms? The reason is that the bases lie in parallel planes. The segments joining them (the sides of the lateral faces), are parallel to each other, and the sides of the congruent polygons are parallel to each other. A pair of segments and a pair of sides make up the sides of the lateral faces, so each lateral face is a parallelogram.

One special kind of prism is a right prism. In a right prism, the lateral faces are all rectangles, and the lateral edges are perpendicular to the planes that contain the bases. One example of a right prism is a cube. A cube is a six-sided polyhedron whose faces are all congruent squares. Below a right prism is drawn:

Cylinders

Prisms are only one member in a larger group of geometric surfaces. That larger group is the set of cylinders. A cylinder is a surface that consists of two congruent simple closed curves lying in parallel planes and the segments that connect them. If these simple closed curves were polygons, then the cylinder would be a prism. Here is a drawing of a cylinder.

We've already talked about cylinders whose bases are polygons. Another kind of cylinder with a special base is a circular cylinder. As you may have already guessed, a circular cylinder is a cylinder with circular bases. In addition to that, a right circular cylinder is a circular cylinder whose lateral surface contains segments that are perpendicular to the bases. A right circular cylinder is drawn below.

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A polyhedron is a geometric solid in three dimensions with flat faces and straight edges.

Traingular Prism A Regular Polyhedron

Polyhedral Surface

A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions.

Edges

Edges have two important characteristics (unless the polyhedron is complex):

• An edge joins just two vertices.
• An edge joins just two faces.

These two characteristics are dual to each other.

Common Polyhedra

	Platonic Solids
	Prisms
	Pyramids

Convex Polyhedrons

The idea of convex polyhedron is similar to that of convex polygons. 

These are convex polyhedrons

These are not convex polyhedrons

Regular Polyhedrons

A polyhedron is said to be regular if its faces are made up of regular polygons and the same number of faces meet at each vertex. 

1. This polyhedron is regular. 2. Its faces are congruent, regular polygons. Vertices are formed by the same number of faces

1. This polyhedron is not regular. 2. All the sides are congruent; but the vertices are not formed by the same number of faces. 3 faces meet at A but 4 faces meet at B.

Two important members of polyhedron family around are prisms and pyramids.

These are prisms

These are pyramids

Prism

A solid whose two faces are parallel plane polygons and the side faces are rectangles is called a prism. A solid whose base and top are identical polygons and the sides are rectangles, is known as a prism. It is a polyhedron, two of whose faces are congruent polygons in parallel planes and whose other faces are parallelograms.

TYPES	FIGURE	FACES	EDGES	VERTICES
(i)	Triangular Prism		5	9	6
(ii)	Cuboid Rectangular Prism		6	12	8
(iii)	Square Prism		6	12	8
(iv)	Cube		6	12	8
(v)	Pentagonal Prism		7	15	10
(vi)	Cylinder		3	2	–

Pyramids

A pyramid is a polyhedron whose base is a polygon (of any number of sides) and whose other faces are triangles with a common vertex.

TYPES	FIGURE	FACES	EDGES	VERTICES
(i)	Triangular Pyramid		4	6	4
(ii)	Rectangular Pyramid		5	8	5
(iii)	Square Pyramid		5	8	5
(iv)	Pentagonal Pyramid		6	10	6

• A prism is a polyhedron whose base and top are congruent polygons and whose other faces, i.e., lateral faces are parallelograms in shape. 
• A pyramid is a polyhedron whose base is a polygon (of any number of sides) and whose lateral faces are triangles with a common vertex.
• A prism or a pyramid is named after its base. Thus a hexagonal prism has a hexagon as its base; and a triangular pyramid has a triangle as its base.

Any polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions:

3 dimensions: The body is bounded by the faces, and is usually the volume enclosed by them.

2 dimensions: A face is a polygon bounded by a circuit of edges, and usually including the flat (plane) region inside the boundary. These polygonal faces together make up the polyhedral surface.

1 dimension: An edge joins one vertex to another and one face to another, and is usually a line segment. The edges together make up the polyhedral skeleton.

0 dimensions: A vertex (plural vertices) is a corner point.

-1 dimension: The null polytope is a kind of non-entity required by abstract theories.

EULER’S Formula

The table below shows the number of faces, edges and vertices of each of the platonic solids. Here, v stands for vertices, f for faces and e for edges.

The above table clearly shows that

F + V = E + 2

Leonard Euler (1707-1783) discovered this formula which established the relationship among the number of faces, edges and vertices of a polyhedron.

Euler’s Formula

F + V = E + 2

Where F = number of faces

V = number of vertices

E = number of edges.

e.g. Try it on the cube:

A cube has 6 Faces, 8 Vertices, and 12 Edges,

so: 6 + 8 - 12 = 2

EULER CHARACTERISTIC:

The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:

x = V – E + F

For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected and whose boundary is a manifold, χ = 2.

So, F+V-E can equal 2, or 1, and maybe other values, so the more general formula is

F + V - E = χ

Where χ is called the "Euler Characteristic".

Here are a few examples:

Shape		χ
Sphere		2
Torus		0
Mobius Strip		0

And the Euler Characteristic can also be less than zero. This is the "Cubohemioctahedron": It has 10 Faces (it may look like more, but some of the "inside" faces are really just one face), 24 Edges and 12 Vertices, so: F + V - E = -2

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