Volumes
 Cross Sections
 End Area Method
 Prismoidal Formula
 Rectangular Base Method
 Triangular Base Method
 Balance of Cut and Fill
 Volumes From Contours
 Digital Terrain Models

Prismoidal Formula

The Prismoidal formula is sometimes called "Simpson's Rule for Volumes", and the derivation is exactly the same as before (see Areas). It is a modification of the End Areas Formula.

An alternative proof can be seen by considering the figure below. Regardless of the combination of rectangular blocks, wedges or prisms, the volume may be expressed in exactly the same form.

V = d

Consider firstly a rectangular prism. Clearly A₁ = A₂ = A₃. Therefore the volume is,

V = A₁.2d = 2A₁.d = 6A₁d/3, but since A₁ = A₂ = A₃

V = d[ A₁ + 4A₂ + A₃]/3, which is the required generalised form.

Secondly, the wedge as shown below:

In this case A₁ = 2A₂ and A₃ = 0. Therefore the total volume is,

V = 1/2.A₁.2d = A₁.d = d[3A₁]/3

V = d[A₁ + 4A₂ + A₃]/3 which is the required generalised form.

spaced cross sections is given by,

This formula may be applied to an odd number (n) of cross sections evenly spaced (d) along a constant direction.