Regular Frustum Calculator

Name: Regular Frustum - Geometry Calculator
Author: Jürgen Kummer

Calculations at a right regular frustum. This is a regular pyramid with the apex truncated parallel to the base.
Enter both side lengths a and b, height h and number of vertices of the bases. Choose the number of decimal places, then click Calculate.

Formulas:

s = \sqrt{\frac{1}{4} \cdot \cot^{2} (\frac{π}{n}) \cdot {(a - b)}^{2} + h^{2}}

e = \sqrt{\frac{[4 \cdot s^{2} + {(a - b)}^{2}]}{4}}

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

A = L + \frac{n \cdot (a^{2} + b^{2})}{4 \cdot \tan (\frac{π}{n})}

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

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.

The regular frustums possess the same symmetry properties as the corresponding regular pyramids from which they were derived. This is because the section through the pyramid that creates the truncated shape lies as symmetrically along the planes of symmetry and the axis of symmetry as the pyramid base. A regular frustum is therefore mirror-symmetric about every plane that passes through a vertex or edge of the larger base, through the directly opposite vertex or edge of that base, and likewise through corresponding vertices or edges of the smaller base. There are as many planes of symmetry as there are vertices or edges on one base; this number is denoted here as n. These frustums are rotationally symmetric about an axis through the centers of the two bases at an angle of 360 degrees divided by the number of vertices on one base.

The regular frustum is a special case of the general frustum. A special case of the regular frustum is, in turn, the square frustum.

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