What makes a solid a polyhedron

For example, a cube, prism, or pyramid are polyhedrons. Cones, spheres, and cylinders are non-polyhedrons because their sides are not polygons and they have curved surfaces. The plural of a polyhedron is also known as polyhedra. They are classified as prisms , pyramids, and platonic solids. For example, triangular prism, square prism, rectangular pyramid, square pyramid, and cube platonic solid are polyhedrons. Observe the following figure which shows the different kinds of polyhedrons.

The dimensions of a polyhedron are classified as faces, edges, and vertices. There is a relationship between the number of faces, edges, and vertices in a polyhedron. We can represent this relationship as a math formula known as the Euler's Formula. We can also verify if a polyhedron with the given number of parts exists or not.

For example, a cube has 6 faces, 8 vertices corner points and 12 edges. Let us check whether a cube is a polyhedron or not by using the Euler's formula. Hence proved, cube is a polyhedron. Polyhedra are mainly divided into two types — regular polyhedron and irregular polyhedron. Regular Polyhedron A regular polyhedron is also called a platonic solid whose faces are regular polygons and are congruent to each other.

In a regular polyhedron, all the polyhedral angles are equal. There are five regular polyhedrons. The following is the list of five regular polyhedrons. Irregular Polyhedron A polyhedron with irregular polygonal faces that are not congruent to each other, and in which the polyhedral angles are not equal is called an irregular polyhedron.

Convex Polyhedron A convex polyhedron is just like a convex polygon. If a line segment joining any two points on the surface of a polyhedron entirely lies inside the polyhedron, it is called a convex polyhedron. Concave Polyhedron A concave polyhedron is quite similar to a concave polygon. If a line segment joining any two points on the surface of a polyhedron goes outside the polyhedron, it is called a concave polyhedron.

Depending on the number of flat sides of the polygonal-based polyhedron, we can count the number of faces. No, a sphere is not a polyhedron because it has a curved surface, whereas, polyhedrons have only straight and flat surfaces, for example, a cube is a polyhedron. Yes, a prism is a polyhedron.

A prism has flat faces, straight edges, and sharp vertices. How can this be possible? Try wiggling your first piece of paper within the second. To be able to incline it with respect to the fold you have to decrease the angle of the first piece, or increase the angle of the second. Next we'll consider all possibilities for the number of faces meeting at a vertex of a regular polyhedron. For each possibility we actually construct such a polyhedron, a picture of which you can see close by on this page.

Here are the possibilities:. But now things get a little more subtle. We have looked at all possibilities of congruent regular polygons meeting at a vertex of a polyhedron, but how do we know that there isn't another regular polyhedron for some of these cases? For example, why is the cube the only polyhedron for which three squares meet at each vertex?

The rest of this page gives the answer to this question, but the going will be much harder! The values of these numbers for each of the polyhedra are listed in this table:. Our aim now is to show that for any pair of number n and m the values of the other parameters, f , e , and v are determined uniquely. We'll see below that this equations actually holds for all convex polyhedra.

Given m and n the above three equations determine f , e , and v uniquely, and so there are only five possible regular polyhedra. The result E is known as. Now think of the remaining faces of the polyhedron as made of rubber and stretched out on a table.

This will certainly change the shape of the polygons and the angles involved, but it will not alter the number of vertices, edges, and faces. Now we draw diagonals in the stretched faces out of the Polygons.