Regular Scutoid Calculator

Name: Regular Scutoid - Geometry Calculator
Author: Jürgen Kummer

Calculations for a regular scutoid. In its general form, a scutoid is a prismatic body with a truncated corner. This specific scutoid is based on a regular prism from which an isosceles right tetrahedron has been removed at one corner. An isosceles triangle is formed at the site of the removed corner. The two adjacent side faces are special pentagons, specifically, cut rectangles.
Enter the dimensions of the original prism, as well as the lengths of the two new edges created by removing the corner. Choose the number of decimal places, then click Calculate.

Formulas:

A_{g} = \frac{n \cdot a^{2}}{4 \cdot \tan (\frac{π}{n})}

p = n \cdot a

d = \sqrt{b^{2} - \frac{c^{2}}{4 \cdot \cos^{2} (\frac{π}{n})}}

e = \frac{c}{2 \cdot \cos (\frac{π}{n})}

A = 2 \cdot A_{g} + h \cdot p - \frac{c^{2}}{4} \cdot \tan (\frac{π}{n}) - \frac{c \cdot d}{2 \cdot \cos (\frac{π}{n})} + \frac{c}{4} \cdot \sqrt{4 \cdot b^{2} - c^{2}}

V = A_{g} \cdot h - \frac{c^{2}}{12} \cdot d \cdot \tan (\frac{π}{n})

Lengths and heights 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.

In the formulas, A_g and p refer to the base area and the perimeter, respectively, of the original regular prism. Edge c of the scutoid corresponds to edge c of the removed tetrahedron. Likewise, the removed vertical edge d corresponds to edge d of that tetrahedron. However, the slanted edge b corresponds to edge a of the tetrahedron, and the removed edge e at the base corresponds to edge b of the tetrahedron. The formula for the surface area is calculated based on the prism's surface area, minus the removed faces, plus the newly created triangular face. The volume is simply that of the prism minus that of the tetrahedron.

The shape of the general scutoid emerged from biological studies. Cell layers can develop such structures. The term scutoid is derived from the Latin scutum, meaning shield. The dorsal shields of certain beetles exhibit a shape, an isosceles triangle situated between adjacent surfaces, that resembles a scutoid. Furthermore, the name of head of the research group at the University of Seville investigating these structures is Escudero, which likely inspired the name scutoid.

Calculation of volume and other dimensions: regular scutoid

Last updated on 08/19/2026. Author: Jürgen Kummer

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