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Let's suppose we have a cuboid $ABCDEFGH$ and now we cut the pyramid $ABCF$ of that cuboid.

What is the ratio of volume of the pyramid to the volume of the original cuboid?

Is it $1\over4$ or $1\over3$?

dfeuer
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user1111261
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  • Define "cuboid". Under Wikipedia's definition, the ratio could be anything - just move points $D$, $E$, $G$, and $H$ around to change the volume of the cuboid but leave the pyramid's volume unchanged. – Caleb Stanford Sep 08 '13 at 22:13
  • in French: un prisme rectangulaire droit, un cuboïde carré, une boîte carrée, un prisme carré droit, un prisme carré; in Spanish: un Ortoedro; in German: ein Quader – user1111261 Sep 08 '13 at 23:51

1 Answers1

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Your subsequent comment indicates that you’re talking about a cuboid in the narrower sense, i.e., a right (or rectangular) cuboid. Let the base dimensions of the cuboid be $a$ and $b$, and let $c$ be its height. The pyramid has as base the $\triangle ABC$, whose area is $\frac12ab$, and height $c$. Its volume is one-third that of the corresponding cylinder, so its volume is $$\frac13\cdot\frac12abc=\frac16abc\;,$$ $\frac16$ the volume of the cuboid.

This image and the first one on this page show a decomposition of a cube into six pyramids of this type.

Brian M. Scott
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