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Pentagonal Icositetrahedron Calculator

Calculations at a pentagonal icositetrahedron, the dual body of the snub cube. Its faces are axial-symmetric pentagons with the top angle acos(2-t)=80.7517°, base and middle sides b and top sides a. Of this polyhedron, there are two forms which are mirror images of each other, but otherwise identical.
Enter one value and choose the number of decimal places. Then click Calculate.


Eugène Charles Catalan, by Emile Delperée Long edge (a): Left Pentagonal Icositetrahedron
Left pentagonal icositetrahedron
Right Pentagonal Icositetrahedron
Right pentagonal icositetrahedron

24 faces, 60 edges, 38 vertices
Short edge (b):
Snub cube edge (d):
Surface area (A):
Volume (V):
Midsphere radius (rm):
Insphere radius (ri):
Surface-to-volume ratio (A/V):
Round to    decimal places.




Formulas:
b=as+1
d=2a2+2s
A=24a2⋅(2+3s)1+2s⋅1-s1+s
V=4a3⋅(2+3s)⋅1-2s(1+s)⋅(1-4s2)
rm=a2⋅(1+s)⋅(1-2s)
ri=a2⋅(1-2s)⋅(1-s2)

Tribonacci constant t = 1+19+3333+19-33333=1.839286755214161...
s = t-12=0.41964337760708...

The pentagonal icositetrahedron is a Catalan solid. Edge lengths and radiuses have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit -1.

The term pentagonal icositetrahedron derives from Ancient Greek and means a solid with twenty and four faces with five vertices each. As is the case with its dual solid, the snub cube, this figure is chiral. So there exists both a left-handed and a right-handed version of it. Furthermore, just as with the snub cube, the Tribonacci constant appears in the formulas used for its calculation.
Due to its relatively low number of faces, the pentagonal icositetrahedron appears less rounded than the other Catalan solids which are usually listed in the latter section. The ordering of the Catalan solids commonly follows the number of faces found on their corresponding Archimedean solids, not the Catalan solids themselves.

The pentagonal icositetrahedron and the pentagonal hexecontahedron are the only two Catalan solids featuring pentagonal faces. This represents the highest number of vertices per face among these solids. The pentagonal icositetrahedron is rarely found in natural forms, such as crystals. For instance, the mineral cuprite can crystallize that way, but also occurs in other crystal forms. This shape is occasionally utilized in art and design.


Calculation of volume and other dimensions: pentagonal icositetrahedron

Last updated on 06/03/2026.

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Cite this page: Rechneronline (2026) - Pentagonal Icositetrahedron.
Retrieved on 2026-09-10 from https://rechneronline.de/pi/pentagonal-icositetrahedron.php




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