Deltoidal Hexecontahedron Calculator

Calculations at a deltoidal hexecontahedron, the dual body of the rhombicosidodecahedron. A deltoidal hexecontahedron is a polyhedron with equal deltoid or kite faces, those have two angles with 86.97°, one angle with 118.3° and one with 67.8°.
Enter one value and choose the number of decimal places. Then click Calculate.

	Long edge (a):			60 faces, 120 edges, 62 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: $$b=a\cdot \frac{3}{22}\cdot (7-\sqrt{5})$$ $$e=3a\cdot \sqrt{\frac{5-\sqrt{5}}{20}}$$ $$f=\frac{a}{11}\cdot \sqrt{\frac{470+156\sqrt{5}}{5}}$$ $$A=\frac{9}{11}\cdot a^2\cdot \sqrt{10\cdot (157+31\sqrt{5})}$$ $$V=\frac{45}{11}\cdot a^3\cdot \sqrt{\frac{370+164\sqrt{5}}{25}}$$ $$r_m=\frac{3}{20}\cdot a\cdot (5+3\sqrt{5})$$ $$r_i=\frac{3}{2}\cdot a\cdot \sqrt{\frac{135+59\sqrt{5}}{205}}$$ $$\frac{A}{V}=\frac{\frac{9}{45}\cdot \sqrt{10\cdot (157+31\sqrt{5})}}{a\cdot \sqrt{\frac{370+164\sqrt{5}}{25}}}$$

The deltoidal hexecontahedron 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 hexecontahedron is derived from Ancient Greek, hexaconta meaning sixty, indicating a solid with sixty deltoids as its faces. This shape has twenty vertices where three edges meet, thirty vertices where four edges meet, and twelve vertices where five edges meet. These vertices correspond to the equilateral triangles, squares, and regular pentagons of its dual solid, the rhombicosidodecahedron. Due to its large number of identical faces, the deltoidal hexecontahedron is one of the classical polyhedra that most closely resembles a sphere. The pentagonal hexecontahedron also has 60 faces, though these are pentagonal. Only the hexakis icosahedron has even more faces of the Catalan solids, specifically twice as many, which are triangular. The deltoidal hexecontahedron belongs to the same symmetry group as the dodecahedron and the icosahedron.
There are two Catalan solids with kites as faces: the deltoidal hexecontahedron and the deltoidal icositetrahedron.

The deltoidal hexecontahedron does not occur as a crystal form in minerals. Outside of polyhedral models and mathematical art, this shape is also very rarely used in art, design, and architecture.

Last updated on 06/08/2026. Author: Jürgen Kummer

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