Antiprism

A Semiregular Polyhedron constructed with 2 $n$-gons and $2n$ Triangles. The 3-antiprism is simply the Octahedron. The Duals are the Trapezohedra. The Surface Area of a $n$-gonal antiprism is

$$S = 2A_{n{\rm -gon}}+2nA_\Delta$$

$$= 2\left[{{\textstyle{1\over 4}}na^2\cot\left({\pi\over n}\right)}\right]+2n({\textstyle{1\over 4}}\sqrt{3}\, a^2)$$

$$= {\textstyle{1\over 2}}na^2\left[{\cot\left({\pi\over n}\right)+\sqrt{3}\,}\right].$$

See also Octahedron, Prism, Prismoid, Trapezohedron

References

Ball, W. W. R. and Coxeter, H. S. M. ``Polyhedra.'' Ch. 5 in Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 130, 1987.

Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 85-86, 1997.

Weisstein, E. W. ``SolidGeometry.'' Mathematica notebook SolidGeometry.m.