Prismoidal Formula

Theorem

The Prismoidal Formula is a formula for calculating the volume $V$ of a prismatoid $\PP$.

It comes in the following forms:

Formulation 1

$V = \dfrac {h \paren {B + 3 A} } 4$

where:

$h$ is the altitude of $\PP$

$B$ is the area of one of the bases of $\PP$

$A$ is the area of a cross-section of $\PP$ parallel to the bases of $\PP$ at $\dfrac 2 3$ of the distance to the other base of $\PP$.

Formulation 2

$V = \dfrac {h \paren {B_1 + B_2 + 4 A_m} } 6$

where:

$h$ is the altitude of $\PP$

$B_1$ and $B_2$ are the areas of the bases of $\PP$

$A_m$ is the area of a cross-section of $\PP$ parallel to the bases of $\PP$ midway between the bases of $\PP$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): prismoidal formula
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): prismoidal formula