Deltoidal Hexecontahedron calculator and formulas

This function calculates various parameters of a deltoidal hexecontahedron. Entering one value is sufficient for the calculation; all others are calculated from it.

The deltoidal hexacontahedron is a convex polyhedron composed of 60 deltoids. It is dual to the rhombicicosidodecahedron and has 62 vertices and 120 edges.

To perform the calculation select the property you know from the menu and enter its value. Then click on the 'Calculate' button.

Deltoidal hexecontahedron

Eingabe

Argument type

Argument value

Decimal places

Results

Long edge a

short edge b

Short diagonal e

Long diagonal f

Surface A

Volume V

Midsphere radius R_K

Insphere radius R_I

Formulas for the deltoidal hexacontahedron

Surface

\(\displaystyle A=\frac{9}{11}· a^2·\sqrt{10·(157+31·\sqrt{5})}\) \(\displaystyle \;\;\;\; ≈a^2·38.92\)

Volume

\(\displaystyle V=\frac{45}{11}· a^3·\sqrt{\frac{370+164·\sqrt{5}}{25}}\) \(\displaystyle \;\;\;\; ≈a^3·22.21\)

Midsphere radius

\(\displaystyle R_K=\frac{3}{20}· a·(5+3·\sqrt{5})\) \(\displaystyle \;\;\;\; ≈a·1.756\)

Insphere radius

\(\displaystyle R_I=\frac{3}{2}· a·\sqrt{\frac{135+59·\sqrt{5}}{205}}\) \(\displaystyle \;\;\;\; ≈a·1.712\)

Sizes of the kite square

Long edge

\(\displaystyle a=\frac{22·b}{3·(7-\sqrt{5})}\) \(\displaystyle \;\;\;\; ≈b·1.54\)

Short edge

\(\displaystyle b=\frac{3· a·(7-\sqrt{5})}{22}\) \(\displaystyle \;\;\;\; ≈\frac{a}{1.54}\)

Short diagonal

\(\displaystyle e= 3·a ·\sqrt{\frac{5-\sqrt{5}}{20}}\) \(\displaystyle \;\;\;\; ≈a· 1.115\)

Long diagonal

\(\displaystyle f=\frac{a}{11} ·\sqrt{\frac{470+156·\sqrt{5}}{5}}\) \(\displaystyle \;\;\;\; ≈a· 1.163\)

Side angle

\(\displaystyle cos\;\; \alpha=\frac{1}{10}·(5-2·\sqrt{5})\) \(\displaystyle \;\; ≈86°\;58'\;27''\)

Base angle

\(\displaystyle cos\;\; \beta=\frac{1}{40}·(9·\sqrt{5}-5)\) \(\displaystyle \;\; ≈67°\;46'\;59''\)

Head angle

\(\displaystyle cos\;\; \gamma=-\frac{1}{20}·(5+2·\sqrt{5})\) \(\displaystyle \;\; ≈118°\;16'\;7''\)

Deltoidal Hexecontahedron

Calculate deltoidal hexacontahedron

Formulas for the deltoidal hexacontahedron

Sizes of the kite square

Catalan solid functions

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