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

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opacity
opacity blue
opacity green
A prismatoid is the convex hull of two parallel convex polygons, a base
B
and a top
T
with areas
S
B
and
S
T
. Suppose the height of the solid is
h
. Make a cross section
M
at height
h/2
, let the point O be on
M
, and let the area of
M
be
S
M
. The sum of the volumes of the two blue pyramids, the first with apex O and base
B
and the other with apex O and base
T
, is
(
S
B
+
S
T
)h/6
. A green pyramid
P
i
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
A
i
h/6
, where
A
i
is the area of
P
i
⋂M
. The sum of all such areas
S
i
is
S
M
. So the volume of the prismatoid is
(
S
B
+
S
T
+4
S
M
)h/6
.
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