Prismatoid and Prismoid Calculator

Name: Prismatoid and Prismoid - Geometry Calculator
Author: Jürgen Kummer

Calculations at a prismatoid. A prismatoid is a polyhedron with two parallel polygons as base and cover faces, whose vertices are connected in a way, that the side faces are triangles or trapezoids (including parallelograms). If both parallel polygons have the same number of vertices and if the side faces all are trapezoids, this is a prismoid. The calculation of the volume is made with the height, the areas of base and cover face and the area of the cross section at half height. The difficulty often lies in determining the cross-sectional area, but there is an approximation formula for this.
Enter four values and choose the number of decimal places, then click Calculate. For the approximation of A₂, please enter A₁ and A₃.

Formula:

V = \frac{h}{6} \cdot (A_{1} + 4 A_{2} + A_{3})

This is a variation of the barrel formula of Kepler.

A_{2} \approx {(\frac{\sqrt{A_{1}} + \sqrt{A_{3}}}{2})}^{2}

Approximation formula for the cross section area.

The height has a one-dimensional unit (e.g. meter), the areas have this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter).

The surface area of such a figure can be calculated from the individual areas of its faces and its base and top surfaces. It is simply the sum of these individual areas. Similarly, the cross-sectional area could also be calculated exactly if it is possible to define the corresponding polygon precisely. Using the approximation above is, of course, much easier. It yields very good values when the base and top surfaces are as similar as possible. Similarity means that the shape is the same, but not necessarily the size. Therefore, if the prismoid is a frustum, this approximation formula provides the exact value for the cross-sectional area.

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