Pyramid

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A pyramid is a polyhedron with one face (known as the "base") a polygon and all the other faces triangles meeting at a common polygon vertex (known as the "apex"). A right pyramid is a pyramid for which the line joining the centroid of the base and the apex is perpendicular to the base. A regular pyramid is a right pyramid whose base is a regular polygon. An n-gonal regular pyramid (denoted Y_n) having equilateral triangles as sides is possible only for n=3, 4, 5. These correspond to the regular tetrahedron, square pyramid, and pentagonal pyramid, respectively.

Canonical n-pyramids are illustrated above for n=3 to 7.

The illustration above shows canonical n-pyramids together with their duals. As can be seen, such pyramids are self-dual, corresponding to the fact that a pyramid's skeleton (a wheel graph) is a self-dual graph. Canonical n-pyramids with unit midradius and midcenter at the origin have regular polygon base circumradius

and base and apex at heights

giving an overall height

The corresponding edges lengths, generalized diameter, circumradius, surface area, and volume are

Nets for the canonical n-dipyramids for n=3, 4, ..., 10 are illustrated above. The faces of the canonical n-pyramid are isosceles triangles with angles

An arbitrary pyramid has a single cross-sectional shape whose lengths scale linearly with height. Therefore, the area of a cross section scales quadratically with height, decreasing from A_b at the base (z=0) to 0 at the apex (assumed to lie at a height z=h). The area at a height z above the base is therefore given by

As a result, the volume of a pyramid, regardless of base shape or position of the apex relative to the base, is given by

Note that this formula also holds for the cone, elliptic cone, etc.

The volume of a pyramid whose base is a regular n-sided polygon with side a is therefore

Expressing in terms of the circumradius of the base gives

(Lo Bello 1988, Gearhart and Schulz 1990).

The geometric centroid is the same as for the cone, given by

The lateral surface area of a pyramid is

where s is the slant height and p is the base perimeter.

Joining two pyramids together at their bases gives a dipyramid, also called a bipyramid.

See also

Augmentation, Dipyramid, Elevatum, Elongated Pyramid, Gyroelongated Pyramid, Hexagonal Pyramid, Invaginatum, Pentagonal Pyramid, Pyramidal Frustum, Square Pyramid, Tetrahedron, Triangular Pyramid, Truncated Square Pyramid 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. 128, 1987.Gearhart, W. B. and Schulz, H. S. "The Function sinx/x." College Math. J. 21, 90-99, 1990.Harris, J. W. and Stocker, H. "Pyramid." §4.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 98-99, 1998.Hart, G. "Pyramids, Dipyramids, and Trapezohedra." http://www.georgehart.com/virtual-polyhedra/pyramids-info.html.Kern, W. F. and Bland, J. R. "Pyramid" and "Regular Pyramid." §20-21 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 50-53, 1948.Lo Bello, A. J. "Volumes and Centroids of Some Famous Domes." Math. Mag. 61, 164-170, 1988.

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Pyramid

Cite this as:

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

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