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
, where
is the area of
. The sum of all such areas
is
. So the volume of the prismatoid is
.
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