Hollow Frustum Calculator

Name: Hollow Frustum - Geometry Calculator
Author: Jürgen Kummer

Calculations at a hollow regular frustum. This is a frustum from which a smaller frustum has been removed from its center. The removed frustum has sides that are shorter by the same value at the large and at the small bases as the larger frustum. This creates a hollow frustum with a constant wall thickness.
Enter the number of vertices of the bases, one of the long and short sides each, as well as the thickness and the height. Choose the number of decimal places, then click Calculate.

Formulas:

a = a' + 2 d

b = b' + 2 d

A = \frac{n}{4} \cdot (a + b) \cdot \sqrt{\cot^{2} (\frac{π}{n}) \cdot {(a - b)}^{2} + 4 h^{2}} + \frac{n}{4} \cdot (a^{'} + b^{'}) \cdot \sqrt{\cot^{2} (\frac{π}{n}) \cdot {(a^{'} - b^{'})}^{2} + 4 h^{2}} + \frac{n \cdot (a^{2} - {a'}^{2})}{4 \cdot \tan (\frac{π}{n})} + \frac{n \cdot (b^{2} - {b'}^{2})}{4 \cdot \tan (\frac{π}{n})}

V = \frac{h}{3} \cdot {n \cdot \frac{a^{2} + b^{2}}{4 \cdot \tan (\frac{π}{n})} + \sqrt{n^{2} \cdot a^{2} \cdot b^{2} / {[4 \cdot \tan (\frac{π}{n})]}^{2}} - n \cdot \frac{{a'}^{2} + {b'}^{2}}{4 \cdot \tan (\frac{π}{n})} - \sqrt{n^{2} \cdot {a'}^{2} \cdot {b'}^{2} / {[4 \cdot \tan (\frac{π}{n})]}^{2}}}

Lengths, thickness 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 regular hollow frustum has exactly as many planes of symmetry, to which it is mirror-symmetric, as vertices at each base. With an even number, there are two types of symmetry planes. One type passes through four opposite vertices at the large base and through four more vertices directly above them at the small base. The second type also passes through the centers of sides a and a', as well as b and b'. With an odd number of vertices, these two types coincide, and the planes of symmetry pass through the vertices and the centers of the opposite sides. The hollow frustum is also rotationally symmetric at an angle of 360 degrees divided by the number of vertices; the axis of rotation passes through the centers of the bases from top to bottom through both truncated pyramids. This axis therefore does not intersect or touch the hollow frustum.
The volume of the hollow frustum is of course calculated as the volume of the outer frustum minus the volume of the inner frustum. The surface area is the volume of the outer frustum plus the lateral surfaces of the inner frustum minus the base surfaces of the inner frustum.

A much simpler shape to calculate, with straight edges and a hole, is the hollow cuboid.

Calculation of volume and other dimensions: hollow frustum

Last updated on 07/26/2026. Author: Jürgen Kummer

© Jumk.de Webprojects | Online Calculators

↑ up