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Pentagonal Pyramid


A pentagonal pyramid is pyramid having a pentagonal base. The edge length e and slant height s of a pentagonal pyramid with regular base of side length a are given by

e=sqrt(h^2+1/(10)(5+sqrt(5))a^2)
(1)
s=sqrt(h^2+1/(20)(5+2sqrt(5))a^2),
(2)

where h is the height and a is the length of a side of the base. It has surface area and volume

S=(5a(a+sqrt(a^2+4(5-2sqrt(5))h^2)))/(4sqrt(5-2sqrt(5)))
(3)
V=1/(12)a^2hsqrt(25+10sqrt(5)).
(4)
J02
J02Net

The regular pentagonal pyramid having equilateral triangles as faces so that all its edges are of the same length is Johnson solid J_2.

For the equilateral pentagonal pyramid with edge length a, the slant height is

 s=1/2sqrt(3)a,
(5)

and the surface area and volume are

S=1/2a^2sqrt(5/2(10+sqrt(5)+sqrt(75+30sqrt(5))))
(6)
V=1/(24)(5+sqrt(5))a^3.
(7)

See also

Johnson Solid, Pentagon, Pentagonal Prism, Pyramid, Regular Pyramid, Square Pyramid, Triangular Pyramid

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Cite this as:

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

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