Prismoidal formula for volume

A prismoid is a solid having two parallel end faces (the two end sections, areas $A_1$ and $A_2$) joined by plane or ruled surfaces. The earthwork between two cross-sections is generally treated as a prismoid. The prismoidal formula gives its volume accurately.

Derivation

Let $L$ be the distance between the end sections, and let $x$ be the distance measured from the first end. Since each linear dimension of the section varies linearly with $x$, the cross-sectional area at distance $x$ is a second-degree (quadratic) function of $x$:

$$A(x) = a + b x + c x^2$$

The areas at the ends and at the middle are

$$A_1 = A(0) = a,\quad A_m = A\left(\tfrac{L}{2}\right) = a + \frac{bL}{2} + \frac{cL^2}{4},\quad A_2 = A(L) = a + bL + cL^2$$

The volume is

$$V = \int_0^L A(x)\,dx = aL + \frac{bL^2}{2} + \frac{cL^3}{3}$$

Now $A_1 + 4A_m + A_2 = 6a + 3bL + 2cL^2$, so

$$\frac{L}{6}\left(A_1 + 4A_m + A_2\right) = aL + \frac{bL^2}{2} + \frac{cL^3}{3} = V$$ $$\boxed{V = \frac{L}{6}\left(A_1 + 4A_m + A_2\right)}$$

where $A_m$ is the area of the middle section (not the mean of $A_1$ and $A_2$; its dimensions are the means of the end dimensions).

Prismoidal rule for a number of sections

If there is an odd number of sections at equal spacing $d$ along the line (that is, an even number of intervals), apply the formula to each pair of intervals (length $2d$, with the section between them as the middle section):

$$V = \frac{d}{3}\left[A_1 + A_n + 4(A_2 + A_4 + \dots) + 2(A_3 + A_5 + \dots)\right]$$

This is the same as Simpson's one-third rule. If the number of sections is even, the last interval is treated by the trapezoidal formula.

Prismoidal correction

The trapezoidal (average end area) formula gives $V_t = \frac{L}{2}(A_1 + A_2)$. The difference between the trapezoidal and prismoidal volumes is the prismoidal correction:

$$C_p = V_t - V_p = \frac{L}{3}\left(A_1 + A_2 - 2A_m\right)$$

For a level section with side slope $s:1$ and end depths $h_1$ and $h_2$, this becomes

$$C_p = \frac{s\,L}{6}\,(h_1 - h_2)^2$$

For level sections it is always positive, so the prismoidal volume = trapezoidal volume - $C_p$. The correction is large only where the depths $h_1$ and $h_2$ differ greatly.