Prismatoid
A prismatoid is a geometric body. It is a polyhedron with parallel polygons as the base and top surface as well as triangles , trapezoids or parallelograms as side surfaces. In contrast to the prism , the base and top surface do not have to be congruent or have the same number of corners. A prismatoid in which the base and top surfaces are polygons with the same number of corners and whose side surfaces consist only of trapezoids and parallelograms is also known as a prismoid .
The volume of a prismatoid can be calculated according to the formula
to calculate. Here are the base area, the area at medium height, the roof area and the height.
This is one of the most famous and universal volume formulas. In honor of its discoverer Johannes Kepler , it is called Kepler's barrel rule .
The prismatoids include:
The scutoid is not a prismatoid in the true sense of the definition , as it is not a polyhedron due to its curved boundary surfaces.
literature
- Amos Day Bradley: Prismatoid, Prismoid, Generalized Prismoid. In: The American Mathematical Monthly. Volume 86, No. 6, June / July 1979, pp. 486-490, JSTOR 2320427 .
- Bruce E. Meserve, Robert E. Pingry: Some Notes on the Prismoidal Formula. In: The Mathematics Teacher. Volume 45, No. 4, April 1952, pp. 257-263, JSTOR 27954012 .
- Claudi Alsina, Roger B. Nelsen: A Mathematical Space Odyssey. Solid Geometry in the 21st Century (= The Dolciani Mathematical Expositions. 50). The Mathematical Association of America, Washington DC 2015, ISBN 978-0-88385-358-0 , pp. 85-89 .
Web links
- Eric W. Weisstein : Prismatoid . In: MathWorld (English).
Individual evidence
- ^ Claudi Alsina, Roger B. Nelsen: A Mathematical Space Odyssey. Solid Geometry in the 21st Century (= The Dolciani Mathematical Expositions. 50). The Mathematical Association of America, Washington DC 2015, ISBN 978-0-88385-358-0 , pp. 85-89 .
- ↑ Note: In the German literature there is occasionally no distinction between prismatoid and prismoid and the terms are instead used synonymously, e.g. B. at Nitschke.
- ↑ ^{a } ^{b} Martin Nitschke: Geometry. Application-related basics and examples for engineers. 2nd updated edition. Fachbuchverlag Leipzig in Carl-Hanser-Verlag, Munich 2014, ISBN 978-3-446-44143-9 , pp. 50-51 .