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Geometric Surfaces

Regular Polyhedra and Spheres

Problems

Problems

Regular Polyhedra

Some of the most specialized geometric surfaces are the regular polyhedra. In the special cases we've studied so far, the base or bases of a geometric surface is a special shape. In a regular polyhedron, all of the polygons that compose the polyhedron are special: they are all congruent regular polygons. Only five regular polyhedra exist. Their names and number of faces are as follows:

  1. A tetrahedron has four faces.
  2. A cube has six faces.
  3. An octahedron has eight faces.
  4. A dodecahedron has 12 faces.
  5. An isocahedron has 20 faces.
A few of these regular polyhedra are drawn below.
Figure %: The regular polyhedra
The tetrahedron, octahedron, and icosahedron are composed of congruent triangles. The cube is composed of congruent squares, and the dodecahedron is composed of regular pentagons.

Spheres

Another very specific geometric surface is the sphere. A sphere consists of all the points that are equidistant from a given fixed point in space. This fixed point is the center of the sphere; a segment with one endpoint at the center and one on the sphere is a radius. A sphere is basically like a three-dimensional circle. In a way, it is also like a regular polyhedron with an infinite number of faces, such that the area of each face approaches zero. This limit, however, does not exist because the set of regular polyhedra is finite--a regular polyhedron cannot have more than 20 faces.

Just as a semicircle is a 180 degree arc, or half a circle, a hemisphere is half a sphere. A hemisphere is drawn below.

Figure %: A hemisphere
Spheres are difficult to represent on a two-dimensional computer screen, so to try to visualize a sphere, it may be best to study the hemisphere figure and imagine two hemispheres joined together. There are also countless examples of spheres or near-spheres in real life. Basketballs and bowling balls are spherical. So are Earth and the other planets in this solar system. Luckily for geometry students, the terms in which spheres are defined and the rules by which spheres are governed are simple.

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