# Newton-Simpson's Formula for the Volume of a Prismatoid

Newton-Simpson's Formula for the Volume of a Prismatoid

A prismatoid is the convex hull of two parallel convex polygons, a base and a top with areas and . Suppose the height of the solid is . Make a cross section at height , let the point O be on , and let the area of be . The sum of the volumes of the two blue pyramids, the first with apex O and base and the other with apex O and base , is . A green pyramid with apex O and a side face has volume equal to four times its upper part (the tetrahedron with vertices O, 12, 11, 6). But the volume of this tetrahedron is h/6, where is the area of ⋂M. The sum of all such areas is . So the volume of the prismatoid is .

B

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h/2

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M

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M

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(+)h/6

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A

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A

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(++4)h/6

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M