An octahedron is a polyhedron having eight faces. Examples include the augmented
triangular prism (Johnson solid  ), boat, gyrobifastigium (Johnson solid  ), heptagonal
pyramid, hexagonal prism, (regular)
octahedron, square dipyramid,
triangular cupola (Johnson solid  ), tridiminished icosahedron (Johnson
solid  ), and truncated tetrahedron.
There are 257 convex octahedra, corresponding to the duals of the octahedral graphs. The convex octahedra consisting of regular
polygonal faces of equal edge lengths are summarized in the following table. They
all have  , as required by the polyhedral formula.
| polyhedron | degree sequence |  |  | | truncated tetrahedron | 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 | 12 | 18 | | heptagonal pyramid | 3, 3, 3, 3, 3, 3, 3,
7 | 8 | 14 | | triangular cupola | 3, 3, 3, 3, 3, 3, 4, 4, 4 | 9 | 15 | | tridiminished
icosahedron | 3, 3, 3, 3, 3, 3, 4, 4, 4 | 9 | 15 | | gyrobifastigium | 3, 3, 3, 3, 4, 4, 4, 4 | 8 | 14 | | augmented triangular prism | 3, 3, 4, 4, 4, 4, 4 | 7 | 13 | | octahedron | 4, 4, 4, 4, 4, 4 | 6 | 12 |
 The regular octahedron
is the Platonic solid  with six polyhedron vertices, 12 polyhedron edges, and eight equivalent equilateral triangular faces, denoted  . It is also
uniform polyhedron  and Wenninger
model  . It is given by the Schläfli symbol  and Wythoff symbol  . The octahedron
of unit side length is the antiprism
of  sides with height  . The
octahedron is also a square dipyramid
with equal edge lengths.
There are 11 distinct nets for the octahedron, the same as for the cube (Buekenhout and Parker 1998). Questions of polyhedron coloring of the octahedron can be addressed using
the Pólya enumeration
theorem.
The dual polyhedron of an octahedron with unit edge lengths is a cube with
edge lengths  .
The illustration above shows an origami octahedron constructed from a single sheet of paper (Kasahara and Takahama 1987,
pp. 60-61).
Like the cube, it has the  octahedral group of symmetries.
The connectivity of the vertices is given by the octahedral graph.
The octahedron has a single stellation: the stella octangula. The solid
bounded by the two tetrahedra of
the stella octangula (left
figure) is an octahedron (right figure; Ball and Coxeter 1987).
The following table gives polyhedra which can be constructed by cumulation of an octahedron by pyramids of given heights  .
Three orientations of an octahedron are illustrated above. The left one has vertices  ,  ,
 , 
(for edge lengths  ), the middle one has vertices
 ,  ,  (for edge lengths  ), and the
right one has vertices  and  (for
edge lengths  ).
In the former case, the face planes are  ,
so a solid octahedron is given by the equation
 |
(1)
|
If the edges of an octahedron are divided in the golden ratio such that the points of division for any face form an equilateral triangle, then the twelve points of division form
an icosahedron (Wells 1991). In
fact, there are two ways in which the edges can be internally divided in the golden ratio and two ways in which they can be externally divided,
resulting in four possible icosahedra. Keeping the same connectivity, but reversing
the long and short ends of the division gives Jessen's orthogonal icosahedron.
A plane perpendicular to a  axis of an octahedron cuts the solid in a regular
hexagonal cross section (Holden 1991, pp. 22-23). Since there are
four such axes, there are four possible hexagonal cross sections.
The centers of the faces of an octahedron form a cube, and the centers of the faces of a cube
form an octahedron (Steinhaus 1999, pp. 194-195). Faceted forms of the octahedron
include the cuboctatruncated
cuboctahedron and tetrahemihexahedron.
Let an octahedron be length  on a side. The
height of the top polyhedron vertex
from the square plane is also the circumradius
 |
(2)
|
where
 |
(3)
|
is the diagonal length, so
 |
(4)
|
Now compute the inradius.
so
 |
(8)
|
Use similar triangles to obtain
so the inradius is
 |
(12)
|
and twice the inradius gives the height of the octahedron viewed as a 3-sided antiprism.
The midradius of the octahedron is
 |
(13)
|
The area of one face is the area of an equilateral
triangle
 |
(14)
|
The volume is two times the volume of a square-base pyramid,
 |
(15)
|
The dihedral angle is
 |
(16)
|
The octahedron can be built using a Haűy
construction. The Haűy octahedral numbers
 |
(17)
|
give another method for calculating the volume
of the octahedron,
 |
(18)
|
in agreement with the result derived above.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York:
Dover, 1987.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton,
FL: CRC Press, p. 228, 1987.
Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension  ." Disc. Math. 186,
69-94, 1998.
Cundy, H. and Rollett, A. "Octahedron.  ." §3.5.3
in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin
Pub., p. 64, 1989.
Davie, T. "The Octahedron." http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/octahedron.html.
Geometry Technologies. "Octahedron." http://www.scienceu.com/geometry/facts/solids/octa.html.
Harris, J. W. and Stocker, H. "Octahedron." §4.4.4 in Handbook of Mathematics and Computational Science. New
York: Springer-Verlag, p. 100, 1998.
Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.
Kasahara, K. Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan
Publications, p. 204, 1988.
Kasahara, K. and Takahama, T. Origami for the Connoisseur. Tokyo: Japan Publications,
1987.
Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 193-195,
1999.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry.
London: Penguin, p. 163, 1991.
Wenninger, M. J. "The Octahedron." Model 2 in Polyhedron Models. Cambridge, England: Cambridge University
Press, p. 15, 1989.
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