Cavalieri's principle

Cavalieri's principle states that figures of equal height have equal area or volume when corresponding cross-sections at every level have equal measure.

Cross-section: A slice or intersection of a solid or other object with a plane. The principle compares corresponding slices taken parallel to a common base.
Bonaventura Cavalieri: An Italian mathematician whose 1635 work developed a method for comparing geometric magnitudes by indivisibles. His name became attached to the principle in its seventeenth-century formulation.
Volume of a sphere: The three-dimensional measure of the region enclosed by a sphere. A sphere's slices can be matched to those of a cylinder with a conical cavity.
Cavalieri's quadrature formula: A formula for the volume of a solid whose parallel cross-sections have areas following a polynomial in height. Unlike the general comparison principle, this formula calculates a particular class of slice patterns.
Plane (geometry): A flat two-dimensional surface extending infinitely in every direction within three-dimensional space. Parallel planes define the levels at which solid cross-sections are compared.
Fubini's theorem: A theorem allowing a multiple integral to be computed as iterated integrals under suitable conditions. It gives an integral formulation of adding the areas of cross-sections through a solid.
Indivisibles: Infinitely small geometric elements used in early methods for finding areas and volumes. Cavalieri interpreted figures as collections of parallel elements to compare their measures.
Volume of a cone: The space enclosed by a cone, equal to one-third the product of its base area and height. Its shrinking parallel cross-sections make it a standard application of the principle.
Brunn–Minkowski inequality: An inequality relating the volumes of sets to the volume of their Minkowski sum. It bounds volumes of combined shapes instead of proving equality from identical slices.
Parallel lines: Coplanar lines that never meet and remain the same distance apart. The planar form uses families of parallel lines to slice figures at corresponding levels.

Linked from 10 pages