Rotunda Calculator

Calculations at a regular pentagonal rotunda. Such a form emerges, if an icosidodecahedron is bisected along its peripheral edges. A rotunda is similar to a cupola, but has pentagons instead of quadrangles as side faces. The regular pentagonal rotunda is Johnson solid J₆.
Enter one value and choose the number of decimal places. Then click Calculate.

	Edge length (a):			17 faces, 35 edges, 20 vertices Top surface: regular pentagon, Base surface: regular decagon, Side surfaces: 10 equilateral triangles, 5 regular pentagons
	Height (h):
	Circumsphere radius (rc):
	Surface area (A):
	Volume (V):
	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.

Length, height and circumsphere radius 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.

If the two rotundas created by bisecting an icosidodecahedron are reassembled in a twisted manner, such that the triangles are positioned opposite the triangles and the pentagons opposite the pentagons, the resulting shape can be called a pseudo-icosidodecahedron. It shares the same dimensions as the icosidodecahedron.
The rotunda possesses lower symmetry than the icosidodecahedron. It exhibits reflectional symmetry across five distinct planes, each passing through two opposite edges of the decagon and the center of the pentagon situated opposite that decagon. It also displays rotational symmetry for rotations of 72 degrees and multiples thereof about the axis formed by the intersection of these symmetry planes.

Just as there are crystal structures resembling the icosidodecahedron, structures resembling the rotunda also occur when such a crystal fails to form completely. Such rotunda shapes also occasionally serve as models for space-frame structures supporting glass roofs in architecture. However, in an architectural context, the term rotunda usually refers to a building with a circular floor plan.

Calculation of volume and other dimensions: rotunda

Last updated on 06/17/2026. Author: Jürgen Kummer

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