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Explanation:

Cavalieri's Principal says that if two solids have uniform cross sections parallel  the base, then the ratio of the areas of those cross sections is the ratio of the volumes of the solids being sectioned.

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If we say the left-right dimension of the cuboid shown is its "width" and the front-back dimension is its "depth", then the area of a cross section of the cuboid parallel to the top or bottom is "width"×"depth".

Since the base of each triangle in such a cross section is "depth", and its height is (1/2)"width", the area of the triangle in each cross section is ...

  A = (1/2)bh = (1/2)("depth")((1/2)"width") = 1/4("width"×"depth")

That is, the area of the triangle in each cross section is 1/4 of the are of the cuboid in each cross section. Hence, by Cavalieri's Principle, the volume of the triangular prism is 1/4 the volume of the cuboid.

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