A regular polygon is a polygon with all sides
congruent and all angles congruent such as
equilateral
triangle,
square,
regular
pentagon, regular hexagon, …

By a (convex) regular polyhedron we mean a
polyhedron with the properties that
All its faces are congruent regular polygons.
The arrangements of polygons about the vertices
are all alike.

The regular polyhedra are the bestknown polyhedra that have connected
numerous disciplines such as astronomy,
philosophy, and art through the
centuries.
They are known as the Platonic solids.

Regular Convex (Platonic) Solids
Tetrahedron
(Triangular Pyramid)
5
Source: Wikapedia

Regular Convex (Platonic) Solids
Hexahedron
(Cube)
6

Regular Convex (Platonic) Solids
Octahedron
7

Regular Convex (Platonic) Solids
Dodecahedron
8

Regular Convex (Platonic) Solids
Icosahedron
9

Platonic Solids
~There are only five platonic solids~
Cube
Tetrahedron
Octahedron
Icosahedron
Dodecahedron

Platonic solids were known to humans much
earlier than the time of Plato. There are carved
stones (dated approximately 2000 BC) that have
been discovered in Scotland. Some of them are
carved with lines corresponding to the edges of
regular polyhedra.

Icosahedral dice were used by the ancient
Egyptians.

Evidence shows that Pythagoreans knew about
the regular solids of cube, tetrahedron, and
dodecahedron. A later Greek mathematician,
Theatetus (415 - 369 BC) has been credited for
developing a general theory of regular
polyhedra and adding the octahedron and
icosahedron to solids that were known earlier.

The name “Platonic solids” for regular polyhedra
comes from the Greek philosopher Plato (427 - 347
BC) who associated them with the “elements” and the
cosmos in his book Timaeus.
“Elements,” in ancient beliefs, were the four objects
that constructed the physical world; these elements
are fire, air, earth, and water. Plato suggested that
the geometric forms of the smallest particles of these
elements are regular polyhedra.
Fire is represented by the tetrahedron, earth the
octahedron, water the icosahedron, and the almostspherical dodecahedron the universe.

Number of
Triangles
About each
Vertex
3
Number of Faces
(F)
Number of
Edges
(E)
Number of
Vertices
(V)
Euler
Formula
V+F=E+2

Number of
Triangles
About each
Vertex
3
Number of Faces
(F)
4
Number of
Edges
(E)
Number of
Vertices
(V)
Euler
Formula
V+F=E+2
6
4
4+4=6+2

Platonic Solids
Tetrahedron

Number of
Triangles
About each
Vertex
3
Number of Faces
(F)
4
Number of
Edges
(E)
Number of
Vertices
(V)
Euler
Formula
V+F=E+2
6
4
4+4=6+2
4

Number of
Triangles
About each
Vertex
Number of Faces
(F)
8
Euler
Formula
V+F=E+2
6
4
4+4=6+2
12
6
6+8=12+2
4
Number of
Vertices
(V)
3
Number of
Edges
(E)
4

Platonic Solids
Tetrahedron
Octahedron

Number of
Triangles
About each
Vertex
Number of Faces
(F)
Number of
Edges
(E)
Number of
Vertices
(V)
Euler
Formula
V+F=E+2
6
4
4+4=6+2
12
6
6+8=12+2
3
4
4
8
5

Number of
Triangles
About each
Vertex
Number of Faces
(F)
Number of
Edges
(E)
Number of
Vertices
(V)
Euler
Formula
V+F=E+2
6
4
4+4=6+2
12
6
3
4
8
20
5
4
30
6+8=12+2
12
12+20=30+2

Platonic Solids
Tetrahedron
Octahedron
Icosahedron

Platonic Solids
Cube
Tetrahedron
Octahedron
Icosahedron

Number of
Pentagons
about each
Vertex
Number of Faces
(F)
Number of
Edges
(E)
Number of
Vertices
(V)
Euler
Formula
V+F=E+2
3
12
30
20
20+12=30+2

Platonic Solids
Cube
Tetrahedron
Octahedron
Icosahedron
Dodecahedron

Platonic Solids
~There are only five platonic solids~
Cube
Tetrahedron
Octahedron
Icosahedron
Dodecahedron

We define the dual of a regular polyhedron to
be another regular polyhedron, which is formed
by connecting the centers of the faces of the
original polyhedron

The dual of the tetrahedron is the tetrahedron.
Therefore, the tetrahedron is self-dual.
The dual of the octahedron is the cube.
The dual of the cube is the octahedron.
The dual of the icosahedron is the dodecahedron.
The dual of the dodecahedron is the icosahedron.

Polyhedron
Schläfli Symbol
The
Dual
Number of
Faces
The Shape of Each
Face
Tetrahedron
(3, 3)
(3, 3)
4
Equilateral
Triangle
Hexahedron
(4, 3)
(3,4)
6
Square
Octahedron
(3,4)
(4, 3)
8
Equilateral
Triangle
Dodecahedron
(5, 3)
(3, 5)
12
Regular Pentagon
Icosahedron
(3, 5)
(5, 3)
20
Equilateral
Triangle