Geometric Polyhedra

A polyhedron (a single polyhedra) is a geometric solid object enclosed on all sides by polygons. Every polyhedron has at least four sides or faces each consisting of polygonal flat planes with at least six common straight edges and four vertices which are any points at the tips of the edges that have three or more faces in common. None of the lines in a polyhedron are curvilinear. The smallest polyhedron is a tetrahedron with four faces, tetra meaning the number four and hedron meaning face. An example of a tetrahedron is a pyramid with a triangle base. A hexahedron is any polyhedra with six faces, such as a cube, and an octahedron has eight faces, etc. A box shaped hexahedron with six faces all at 90 degree angles is called a rectangular hexahedron. Another hexahedron is a parallelepiped prism where all six faces are parallelograms. An important group of polyhedra have the same regular polygonal face on all sides and are called regular polyhedra. There are only five known regular polyhedra. The first is the regular tetrahedron pyramid with four equilateral triangle sides. The volume of a regular tetrahedron is the area of the side cubed times the square root of two, all divided by twelve. The next is the regular hexahedron cube with six square sides. The volume of a regular hexahedron is the area of the side cubed. Then there is the regular octahedron with eight equilateral triangle sides. The volume of a regular octahedron is the area of the sides cubed times the square root of two, all divided by three. Fourth is the regular dodecahedron with ten pentagon sides. The volume of a regular dodecahedron is the area of the side cubed times the quantity fifteen plus seven times the square root of five, all divided by four. Finally, there is the regular icosahedron with twenty equilateral triangle sides. The volume of a regular icosahedron is the quantity five times the area of the side cubed divided by twelve times the quantity three plus the square root of five. There are also many semi-regular polyhedra that use more than one regular polygon on all sides. They must also form an enclosed shape where all outer vertices can be inscribed on one sphere.

A cube is a geometric solid where all the six faces are squares and are at right angles. It is a regular polyhedra that is both a rectangular hexahedron and a parallelepiped prism.

Math Mysteries: Each of the five known regular polyhedra complies with the Euler characteristic which states that on any polyhedra that is inscribed inside a sphere the number of vertices plus the number of faces minus the number of edges always equals two. The regular tetrahedron has four vertices, four faces and six edges, 4 + 4 - 6 = 2. The hexahedron cube has eight vertices, six faces and twelve edges, 8 + 6 - 12 = 2. The regular octahedron has six vertices, eight faces and twelve edges, 6 + 8 - 12 = 2. The regular dodecahedron has twenty vertices, ten sides and twenty-eight edges, 20 + 10 - 28 = 2. The regular icosahedron has twelve vertices, twenty faces and thirty edges, 12 + 20 - 30 = 2. It is not known why this characteristic of these solids are true.