Deltoidal Icositetrahedron Calculator

Calculations at a deltoidal icositetrahedron, the dual body of the rhombicuboctahedron. A deltoidal icositetrahedron is a polyhedron with deltoid (kite) faces, those have three angles with 81.579° and one with 115.263°. It has eight vertices with three edges and eighteen vertices with four edges. Hereby, the short sides meet at the corners with three edges, and the long sides meet at the corners with four edges.
Enter one value and choose the number of decimal places. Then click Calculate.

	Long edge (a):			24 faces, 48 edges, 26 vertices Faces: kites (deltoids) One of the deltoid faces.
	Short edge (b):
	Symmetry diagonal (e):
	Other diagonal (f):
	Surface area (A):
	Volume (V):
	Midsphere radius (rm):
	Insphere radius (ri):
	Surface-to-volume ratio (A/V):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.
	Formulas: $$b=\frac{a\cdot (4+\sqrt{2})}{7}$$ $$e=\frac{a}{7}\cdot \sqrt{46+15\sqrt{2}}$$ $$f=\frac{a}{2}\cdot \sqrt{4+2\sqrt{2}}$$ $$A=\frac{12}{7}\cdot a^2\cdot \sqrt{61+38\sqrt{2}}$$ $$V=\frac{2}{7}\cdot a^3\cdot \sqrt{292+206\sqrt{2}}$$ $$r_m=\frac{a}{2}\cdot (1+\sqrt{2})$$ $$r_i=a\cdot \sqrt{\frac{22+15\sqrt{2}}{34}}$$ $$\frac{A}{V}=\frac{6\cdot \sqrt{61+38\sqrt{2}}}{a\cdot \sqrt{292+206\sqrt{2}}}$$

The deltoidal icositetrahedron is a Catalan solid. Edge lengths, diagonals 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 deltoidal icositetrahedron may sound complicated, but it simply means a 24-sided polyhedron composed of kites. This shape can be divided into two halves along three different directions without cutting through any of its original edges. Consequently, it can also be subdivided into quarters and eighths. The planes of division are mutually perpendicular, each producing a cross-section in the shape of a regular octagon with an edge length of a.
Corresponding to the rhombicuboctahedron, there exists a shape with identical faces and volume, the pseudo-rhombicuboctahedron, in which one of the caps is rotated by 45 degrees. The dual form of a pseudo-rhombicuboctahedron is, accordingly, a pseudo-deltoidal icositetrahedron, in which one half of the deltoidal icositetrahedron is rotated by 45 degrees.

Deltoidal icositetrahedra are found in nature. The silicate mineral leucite, also known as white garnet, typically crystallizes in this form. It is also occasionally observed in other minerals of the garnet group and in fluorite.

Last updated on 05/13/2026. Author: Jürgen Kummer

© Jumk.de Webprojects | Online Calculators





↑ up

---