Why are there five platonic solids

The properties of a dodecahedron are:. The properties of an icosahedron are:. Platonic solids are considered to be only 5 solid shapes. Here are the reasons why there are only 5 shapes and not more:. According to Euler's formula, for any convex polyhedron, the Number of Faces plus the Number of Vertices corner points minus the Number of Edges always equals 2. Let us take apply this in one of the platonic solids - Icosahedron.

Example 1: Demi wants to know the name of the platonic solid shown below. Can you name the platonic solid for her? Solution: The given solid has 20 triangular faces and 5 triangles are intersecting at each vertex, which is a property of an icosahedron. Hence the given solid is an icosahedron. Example 2: Rita was given the following information about a platonic solid that it has 3 faces meeting at vertices and has 4 vertices. Can you find out which platonic solid is this? Solution: Out of all platonic solids, only the tetrahedron has 4 vertices.

Hence, the given platonic solid is a tetrahedron. Platonic solids are 3D geometrical shapes with identical faces i. Platonic solids were identified in ancient times and were studies by the ancient greeks. These shapes are also known as regular polyhedra that are convex polyhedra with identical faces made up of congruent convex regular polygons. The 5 platonic solids are considered cosmic solids due to their connection to nature that was discovered by Plato.

The cube represents the earth, the octahedron represents the air, the tetrahedron represents the fire, the icosahedron represents the water, and the dodecahedron represents the universe. There are only 5 platonic solids that exist due to the number of faces, edges, and vertices. It is impossible to have more than 5 platonic solids.

Learn Practice Download. Platonic Solids - Why Five? A Platonic Solid is a 3D shape where: each face is the same regular polygon the same number of polygons meet at each vertex corner There are only five of them Simplest Reason: Angles at a Vertex The simplest reason there are only 5 Platonic Solids is this: At each vertex at least 3 faces meet maybe more.

The faces can be triangles 3 sides , squares 4 sides , etc. Let us call this " s ", the number of s ides each face has. Also, at each corner, how many faces meet? For a cube 3 faces meet at each corner. For an octahedron 4 faces meet at each corner. Let us call this " m " how many faces m eet at a corner.

Those two are actually enough to show what type of solid it is. The most common regular polyhedron is the cube whose faces are congruent squares.

The other regular polyhedra are shown below. Regular polyhedra are also known as Platonic solids — named after the Greek philosopher and mathematician Plato.

The Greeks studied Platonic solids extensively, and they even associated them with the four classic elements: cube for earth, octahedron for air, icosahedron for water, and tetrahedron for fire. Interestingly, even though we can create infinitely many regular polygons, there are only five regular polyhedra. And the proof is fairly easy.

Before we discuss the proof, let us familiarize ourselves with the different terms which we will use in the proof. In the following discussion, vertex will refer to the corner of a Platonic solid, face will refer to the regular polygons that make up the solid, and side edges in 3D will refer the side of the polygon.