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Cavalieri’s principle
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Two geometric solids of equal height are shown side by side, each sliced horizontally at the same levels. Corresponding cross sections are highlighted and connected by arrows, illustrating Cavalieri's principle that solids with equal heights and equal cross-sectional areas at every level have equal volumes.
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Cavalieri’s principle

mathematics
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Two geometric solids of equal height are shown side by side, each sliced horizontally at the same levels. Corresponding cross sections are highlighted and connected by arrows, illustrating Cavalieri's principle that solids with equal heights and equal cross-sectional areas at every level have equal volumes.
Cavalieri's principle Bonaventura Cavalieri observed that figures (solids) of equal height and in which all corresponding cross sections match in length (area) are of equal area (volume). For example, take a regular polygon equal in area to an equilateral triangle; erect a pyramid on the triangle and a conelike figure of the same height on the polygon; cross sections of both figures taken at the same height above the bases are equal; therefore, by Cavalieri's theorem, so are the volumes of the solids.

Cavalieri's principle, developed by mathematician Bonaventura Cavalieri, is a precursor to integral calculus. It provides a method for determining the size of geometric figures by considering them as sums of infinitesimally thin slices, or "indivisibles".12

The principle states that if two solids can be sliced by a family of parallel planes, and each plane cuts both solids into cross sections of equal area, then the two solids have equal volume. This concept was crucial in geometry and helped mathematicians calculate areas and volumes more efficiently before the full development of calculus. Cavalieri's work, particularly his book Geometria Indivisibilibus Continuorum Nova Quadam Ratione Promota, introduced this method, though it initially faced criticism before being refined into a more widely accepted form.21

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2 Britannica Sources
  1. Bonaventura Cavalieri
  2. Archimedes’ Lost Method

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