Pentagonal pyramid

Pentagonal pyramid
Dimensionality	3
Schläfli Symbol	'"`UNIQ--postMath-00000001-QINU`"'
Bowers acronym	peppy
Subfacets
Vertices	6 points
Edges	10 line segments
Faces	5 triangles 1 pentagon
Cells	1 pentagonal pyramid
Hypervolumes
Vertex Count	'"`UNIQ--postMath-00000002-QINU`"'
Edge Length	'"`UNIQ--postMath-00000003-QINU`"'
Surface Area	'"`UNIQ--postMath-00000004-QINU`"'
Surcell Volume	'"`UNIQ--postMath-00000005-QINU`"'

A pentagonal pyramid is a three-dimension polyhedron created by taking the pyramid of a pentagon. This makes it the segmentotope between a pentagon and a point.

Variants

Iscosceles Pentagonal Pyramid

An iscosceles pentagonal pyramid has the same symmetry as the regular polygonal one, except the triangular faces are isosceles instead of regular. When the edge length of the base is a, and the long edges of the triangles have length b, this gives subfacets (along with their accompanying hypervolumes and subfacets) of

• 10 line segments (1D)
  • 5 of length a
  • 5 of length b
• 5 isosceles triangles (2D)
  • edge length = $a + 2b$
  • surface area = $\frac{1}{2}ah$
  • 3 line segments
    • 1 of length a
    • 2 of length b
• 1 pentagon (2D)
  • edge length = $5a$
  • surface area = $B^2$
  • 5 line segments of length a
• 1 iscosceles pentagonal pyramid

The full isosceles pentagonal pyramid itself has hypervolumes of

• edge length = $5a + 5b$
• surface area = $B^2 + \frac{5}{2}ah$
• surcell volume = $B^2 h$,

where $h=\frac{1}{2} \sqrt{4b^2 - a^2 }$, equal to the height of the pyramid, and $B^2 = \frac{5}{4} \sqrt{1 + \frac{2}{ \sqrt{5} } } a^2$, equal to the area of the pentagonal base. When a=b, this becomes the regular polygonal variant and the Johnson solid.

See Also

$() \wedge \{3\}$	$() \wedge \{4\}$	$() \wedge \{5\}$	$() \wedge \{6\}$
Tetrahedron	Square pyramid	Pentagonal pyramid	Hexagonal pyramid