These 5 geometric figures are also known as the 5 Platonic Solids and are the only convex regular polyhedra that can exist.
A regular polyhedron is defined as a solid threedimensional object having faces where
• each face is a regular polygon. (A regular polygon has equal sides and equal angles).
• the same number of faces (or the same number of edges) meet at each vertex
• all the dihedral angles (the angles between the planes) are equal
3 faces and 3 edges meet at each vertex
Each face is an equilateral triangle.
4 faces
4 vertices
6 edges
3 faces and 3 edges meet at each vertex
Each face is a square.
6 faces
8 vertices
12 edges
4 faces and 4 edges meet at each vertex
Each face is an equilateral triangle.
8 faces
6 vertices
12 edges
3 faces and 3 edges meet at each vertex
Each face is a pentagon.
12 faces
20 vertices
30 edges
5 faces and 5 edges meet at each vertex
Each face is an equilateral triangle.
20 faces
12 vertices
30 edges
The Swiss mathematician Leonhard Euler (17071783) discovered the formula
V E +F = 2 which states that the vertices minus the edges plus the faces of a convex polyhedron will always equal two.
If we were to inscribe a sphere within any of the 5 Platonic solids, it would be tangent to the center of each face.
If we were to circumscribe a sphere outside any of the 5 Platonic solids, it would pass through all of the vertices.
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