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

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

.