Commandino's theorem

Commandino's theorem gives a prismatoid's volume as one-sixth its height multiplied by the sum of the areas of its two parallel bases and four times the area of its midsection.

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Prismatoid: A polyhedron whose vertices all lie in two parallel planes. The theorem applies to this solid and its parallel cross-sections.
Polyhedron: A three-dimensional solid bounded by flat polygonal faces. A prismatoid is a polyhedron with vertices restricted to two parallel planes.
Federico Commandino: An Italian mathematician and translator of ancient Greek mathematical works, born in 1509 and died in 1575. The theorem bears his name and is associated with his mathematical work.
Volume: The amount of three-dimensional space occupied by a solid. The theorem computes this quantity from height and section areas.
Prismoidal formula: A surveying formula estimating volume from end areas and a middle cross-section. It is commonly presented as the practical counterpart of Commandino's exact prismatoid result.
Prismatoid formula: A formula for the volume of a prismatoid using its height and cross-sectional areas. This is the general volume relation expressed by Commandino's theorem.
Parallel planes: Distinct planes in three-dimensional space that do not intersect. The theorem's bases lie in parallel planes, defining its height.
Archimedes: An ancient Greek mathematician who established methods for calculating areas and volumes. His work on solids and volume formed part of the tradition surrounding the theorem.
Earthwork calculation: Estimating the volume of soil cut or filled between measured cross-sections. The theorem can calculate volumes of prismatoid-shaped regions between parallel surveys.
Trapezoidal rule: A numerical integration rule estimating an integral from endpoint values. Unlike Commandino's theorem, it uses endpoint data without the midsection's fourfold weight.