Basic Concepts of Polyhedron

Topics: Polyhedron, Polygon, Platonic solid Pages: 7 (1405 words) Published: February 25, 2014
MODULE
BASIC CONCEPTS OF POLYHEDRONS

This module will introduce to you basic ideas about polyhedrons. It will help you determine the surface of polyhedrons. It will also explain to you regular polyhedrons, its classifications and how to construct it.

Learning Goal
This module is written for you to:
1. Define polyhedrons;
2. Identify and illustrate the surface of polyhedrons;
3. Determine convex polyhedrons;
4. Determine regular polyhedrons; and
5. Construct regular polyhedrons.

Let’s do some warm-up

1. What is another name for a corner or a point? _______________

2. The prefix”poly” means _____________.

3. What is another name for cube? _________________

4. How many regular polyhedrons are there? ___________________

Let’s read and understand

Lesson 1
Polyhedrons

A polyhedron (plural polyhedrons or polyhedra) is a three-dimensional geometric solid whose boundary consists of plane polygons. In Greek, poly means “many” and hedron means “face”. The polygons which bound the polyhedron are its faces; the sides of these faces are the edges and their vertices are the vertices of the polyhedron. The faces, edges, and vertices taken together form the surface of the polyhedron.

Examples:
For polyhedrons below, tell the number of vertices, edges, and faces. Also determine the geometric name of the faces.

1.

Answer:
The figure is made up of five triangles and one pentagon so there are six faces. There are six vertices--one on the "top" and five on the "bottom." There are ten edges--five on the "sides," and five on the "bottom."

2.

Answer:
The figure is made up of three triangles only so there are three faces. There are four vertices--one on the "top" and three on the "bottom." There are six edges--three on the "sides," and three on the "bottom."

Let’s have some drill

For polyhedrons below, tell the number of vertices, edges, and faces. Also determine the geometric name of the faces. (3 points each)

Let’s read and understand

Lesson 2
Regular Polyhedrons

A polyhedron whose polyhedral angles are equal and whose faces are equal regular polygons is a regular polyhedron. A polyhedron is named after its number of faces. The only regular polyhedral are tetrahedron (figure 1), which has four triangular faces; the cube (figure 2), which has six square faces; the octahedron (figure 3), which has eight triangular faces; the dodecahedron (figure 4), which has 12 pentagonal faces; and the icosahedrons (figure 5), which has 20 triangular faces. (Note: Figures below are shown in front view only.)

Polyhedrons are sometimes referred to as the platonic solids because they appear in the writing of the Greek philosopher Plato. He equated the tetrahedron with the element ‘fire’, the cube with ‘earth’, the icosahedron with ‘water’, the octahedron with ‘air’, and the dodecahedron with the ‘stuff of which the constellations and heavens were made’ (Cromwell 1997) Polyhedral Angle

A polyhedral angle is an angle formed by the intersection of three or more planes meeting at a point. Given a convex polygon and a point P not in its plane. If a half-line l with its end point fixed at P moves so that it always touches the polygon and is made to traverse it completely, it is said to generate a convex polyhedral angle. The fixed point P is the vertex of the polyhedral angle, and the rays through the vertices of the polygon are the edges of the polyhedral angle. Any two consecutive edges determine a plane, and the portion of such a plane included between these edges is called a face of the polyhedral angle. Polyhedral angle is read by naming the vertex and one letter in each edge, as P - ABCDE, or by naming the vertex alone where no ambiguity would arise.

Face Angles
The plane angles at the vertex are called the face angles of the polyhedral...

References: H. E. SLAUGHT AND N. J. LENNES, The Project Gutenberg EBook of Solid Geometry with Problems and Applications (Revised edition), Release Date: August 26, 2009 [EBook #29807]
WILLIAM JAMES RALPH, Microsoft ® Encarta ® 2009
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