A deltahedron is a polyhedron whose faces are congruent equilateral triangles (Wells 1986, p. 73). Note that polyhedra whose faces could be triangulated so as to be composed of coplanar equilateral triangles sharing an edge (such as the truncated tetrahedron, whose hexagonal faces could be considered as six conjoined equilateral triangles) are not allowed.
The term deltahedron should not be confused with "deltohedron," which is a synonym for trapezohedron.
There are an infinite number of deltahedra, but only eight convex ones (Freudenthal and van der Waerden 1947). The eight convex deltahedra have 
, 6, 8, 10, 12, 14, 16, and 20 faces. These are summarized
in the table below and illustrated in the figures above.
 | name |
| 4 | tetrahedron |
| 6 | triangular dipyramid |
| 8 | octahedron |
| 10 | pentagonal dipyramid |
| 12 | snub disphenoid |
| 14 | triaugmented triangular prism |
| 16 | gyroelongated square dipyramid |
| 20 | icosahedron |
The tritetrahedron and augmentations of the Platonic solids are concave deltahedra,
as is the "caved in" augmented dodecahedron
(icosahedron stellation 
; Wells 1991, p. 78).
Cundy (1952) identified 17 concave deltahedra with two kinds of polyhedron vertices.
