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Regular Pyramid

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A regular pyramid is a right pyramid whose base is a regular polygon. For a right pyramid of height h with a regular n-gonal base of side length a and circumradius R, the lateral edge length is given by

e_n=sqrt(h^2+R^2)=sqrt(h^2+1/4a^2csc^2(pi/n)).
(1)

This gives the special cases

e_3=3/2sqrt(h^2+1/3a^2)
(2)
e_4=2sqrt(h^2+1/2a^2)
(3)
e_5=5/2sqrt(h^2+1/(10)(5+sqrt(5))a^2)
(4)
e_6=3sqrt(h^2+a^2).
(5)

Similarly, the slant height of a regular pyramid with regular n-gonal base of side length a and inradius r is given by

s_n=sqrt(h^2+r^2)
(6)
=sqrt(h^2+1/4a^2cot^2(pi/n)).
(7)

Furthermore, since the perimeter of a regular n-gon of side length a is simply

p_n=na,
(8)

the total surface area is equal to the lateral surface area plus the area of the base,

T_n=A_n+S_n
(9)
=A_n+1/2ps_n
(10)
=A_n+1/2nas_n
(11)
=1/4an[acot(pi/n)+sqrt(4h^2+a^2cot^2(pi/n))].
(12)

See also

Pentagonal Pyramid, Pyramid, Square Pyramid, Triangular Pyramid

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References

Kern, W. F. and Bland, J. R. "Regular Pyramid." §21 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 50-53, 1948.

Referenced on Wolfram|Alpha

Regular Pyramid

Cite this as:

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

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