Truncated Rhombohedron Calculator (Dürer's solid)

The truncated rhombohedron (also truncated triangular trapezohedron) is a polyhedron from Albrecht Dürer's engraving Melencolia I. It is constructed by truncating the two sharp vertices of a rhombohedron with the edge length a and an angular ratio of 2/3 (72° and 108°), so that the remaining solid has a circumscribed sphere, that touches every vertex.
Enter one value and choose the number of decimal places. Then click Calculate.

	Edge length rhombohedron (a):			8 faces, 18 edges, 12 vertices Faces: 6 axial-symmetric pentagons, 2 equilateral triangles with side length c
	Medium edge length (b):
	Edge length triangle (c):
	Circumsphere radius (rc):
	Area of a pentagon (A5):
	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.

Source: de.wikipedia.org/wiki/Rhomboederstumpf, Frankee 67

Length and radius have the same unit (e.g. meter), surface areas have 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 engraving (see image below), which features the truncated rhombohedron, was created in 1514. Since the 20th century, there has been debate regarding the precise nature of this shape. The definition above for a polyhedron with a circumsphere represents the most frequently hypothesized form and arguably the geometrically most interesting one. This hypothesis is supported by other writings in which Dürer explores geometry.

Melencolia I is one of Albrecht Dürer’s three master prints and incorporates numerous enigmatic symbols. It depicts the allegorical figure of Melancholy, who desires to create something great but lacks the drive to do so. This was a state likely familiar to Dürer, yet the creation of this engraving (and many of his other works) also bears witness to the ability to overcome it. Beneath the polyhedron, a sphere is visible as another geometric object. A further mathematical object appearing in the image is a magic square. In such a square, numbers are arranged in an equal number of rows and columns such that the sum of each row and column is identical. Polyhedra resembling the truncated rhombohedron were known prior to Dürer’s time. It is unknown whether this specific truncated rhombohedron was first described by him. Magic squares were likewise already known. The one in Dürer’s image displays the year of its creation, 1514, in the center of the bottom row. We do not know whether this particular magic square was known previously. Alongside mathematics, the image depicts tools used for craftsmanship and creation, as well as objects from the field we now call physics, such as a balance scale and an hourglass.
The original copper printing plate no longer exists, though prints from this time are still held in some museums. However, they are rarely displayed there, as the over 500 years old paper is extremely fragile and must be stored under very specific conditions to protect it from disintegration. The art collections of the City of Nuremberg, Dürer’s hometown, hold one such original print.

Albrecht Dürer, Melencolia I, engraving from 1514.

Last updated on 06/26/2026. Author: Jürgen Kummer

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