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Prism


Prism1Prism2

A general prism is a polyhedron possessing two congruent polygonal faces and with all remaining faces parallelograms (Kern and Bland 1948, p. 28; left figure).

A right prism is a prism in which the top and bottom polygons lie on top of each other so that the vertical polygons connecting their sides are not only parallelograms, but rectangles (right figure). A prism that is not a right prism is known as an oblique prism. If, in addition, the upper and lower bases are rectangles, then the prism is known as a cuboid.

PrismsAndDuals

The regular right equilateral prisms are canonical polyhedra whose duals are canonical dipyramids. Canonical prisms have midradius equal to the circumradius of their n-gonal faces, i.e.,

 rho_n=1/2acsc(pi/n),
(1)

where a is the edge length.

PrismNets

The regular right prisms have particularly simple nets, given by two oppositely-oriented n-gonal bases connected by a ribbon of n squares. The graph corresponding to the skeleton of a prism is known, not surprisingly, as a prism graph.

The volume of a prism of height h and base area A is simply

 V=Ah.
(2)
Prism03Prism04Prism05Prism06
Prism07Prism08Prism09Prism10

The above figure shows the first few regular right prisms, whose faces are regular n-gons. The 4-prism with unit edge lengths is simply the cube. The dual polyhedron of a regular right prism is a dipyramid.

A regular right unit n-prism has surface area

S_n=2A_n+n·1^2
(3)
=n[1+1/2cot(pi/n)],
(4)

where A_n is the area of the corresponding regular polygon. The first few surface areas are

S_3=1/2(6+sqrt(3))
(5)
S_4=6
(6)
S_5=5+1/2sqrt(5(5+2sqrt(5)))
(7)
S_6=3(2+sqrt(3))
(8)
S_7=(64x^6-2688x^5+43120x^4-329280x^3+1181292x^2-1479016x-487403)_6
(9)
S_8=4(3+sqrt(2))
(10)
S_9=(64x^6-3456x^5+66096x^4-513216x^3+918540x^2+6141096x-19309023)_6
(11)
S_(10)=5(2+sqrt(5+2sqrt(5))).
(12)

The algebraic degrees of these areas for n=3, 4, ... are 2, 1, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 8, 4, 16, 6, 18, 4, ... (OEIS A089929).

A regular right unit n-prism has volume

 V_n=1·A_n=1/4ncot(pi/n).
(13)

The first few volumes are

V_3=1/4sqrt(3)
(14)
V_4=1
(15)
V_5=1/4sqrt(5(5+2sqrt(5)))
(16)
V_6=3/2sqrt(3)
(17)
V_7=(4096x^6-62720x^4+115248x^2-16807)_6
(18)
V_8=2(1+sqrt(2))
(19)
V_9=(4096x^6-186624x^4+1154736x^2-177147)_6
(20)
V_(10)=5/2sqrt(5+2sqrt(5)).
(21)

The algebraic degrees of the volumes are the same as for the surface areas.

The right regular triangular prism, square prism (cube), and hexagonal prism are all space-filling polyhedra.


See also

Antiprism, Augmented Hexagonal Prism, Augmented Pentagonal Prism, Augmented Triangular Prism, Biaugmented Pentagonal Prism, Biaugmented Triangular Prism, Cube, Cuboid, Cylinder, Dipyramid, Generalized Cylinder, Hexagonal Prism, Metabiaugmented Hexagonal Prism, Octagonal Prism, Parabiaugmented Hexagonal Prism, Pentagonal Prism, Polygrammic Prism, Prism Graph, Prismatoid, Prismoid, Pyramid, Trapezohedron, Triangular Prism, Triaugmented Hexagonal Prism, Triaugmented Triangular Prism Explore this topic in the MathWorld classroom

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References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 127, 1987.Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 13 and 85-86, 1997.Harris, J. W. and Stocker, H. "Prism." §4.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 96-98, 1998.Kern, W. F. and Bland, J. R. "Prism." §13 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 28-32, 1948.Pedagoguery Software. Poly. http://www.peda.com/poly/.Sloane, N. J. A. Sequence A089929 in "The On-Line Encyclopedia of Integer Sequences."Webb, R. "Prisms, Antiprisms, and their Duals." http://www.software3d.com/Prisms.html.

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Prism

Cite this as:

Weisstein, Eric W. "Prism." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Prism.html

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