Right Deltohedron, Trapezohedron Calculator

Name: Right Deltohedron, Trapezohedron - Geometry Calculator
Author: Jürgen Kummer

Calculations for a right deltohedron or trapezohedron. This is a special form of trapezohedron with right deltoids or kites as faces. Such a solid can also be called a right antidipyramid.
Enter the number of double faces n and the edge length at the equator a. Choose the number of decimal places, then click Calculate.

Formulas:

t = \tan (\frac{π}{n}) \cdot \tan (\frac{π}{2 n})

b = \frac{a}{\sqrt{t}}

h = b \cdot \sqrt{2 + t}

A = 2 n b^{2} \cdot \sqrt{t}

V = \frac{n}{3} b^{3} \cdot \sqrt{t}

Source: Mathematical analysis of trapezohedron/deltohedron having 2n congruent right kite faces Harish Chandra Rajpoot, M.M.M. University of Technology, Gorakhpur-273010 (UP), India, 15 April, 2015

Edge lengths and height 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 t in the formulas above is a dimensionless auxiliary quantity used to simplify them.

A deltohedron has 2n faces, 4n edges and 2n+2 vertices. The shorter equator edges a form the center of this solid, while the longer apex edges b extend from there to one of the two apices. In a right deltohedron, the edges a and b are interdependent. This results from the arrangement of the deltohedra in space to create this shape. This interdependence does not exist in right kites. Therefore, a right trapezohedron cannot be formed from just any right deltoids, but only from those in which the edges are in the ratio described above. A right trapezohedron with two times three faces is a cube.

The terms trapezohedron and deltohedron are generally used synonymously, including here. Trapezohedron is the older term, from a time when trapezoid didn't yet refer to what we now call a trapezoid, but rather to general quadrilaterals. The trapezohedron was therefore originally conceived as an irregular shape. Thus, deltohedron is the more accurate name, but trapezohedron is still more common.

The right trapezohedron has a different edge ratio than the regular trapezohedron. Its apex edges are shorter, therefore the right trapezohedron is flatter than the regular one.

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