The importance of volume assessment has already been outlined. Many volumes encountered in civil engineering appear, at first glance, to be rather complex in shape. Generally speaking, however, they can be divided into prisms, wedges or pyramids, each of which will now be dealt with in turn.
The two ends of the prism are equal and parallel, the resulting sides thus being parallelograms.
Vol = AXL
Volume of wedge
when a = b = c :
Volume of pyramid = AL/3
Equations can all be expressed as the common equation:
where A1 and A2 are the end areas and Am is the area of the section situated mid-way between the end areas. It is important to note that Am is not the arithmetic mean of the end areas, except in the case of a wedge.
To prove the above statement consider:
In this case A1 = Am = A2:
In this case Am is the mean of A1 and A2, but A2 = 0. Thus Am = A/2:
In this case Am = A/4 and A2 = 0:
Thus, any solid which is a combination of the above three forms and having a common value for L, may be solved using equation . Such a volume is called a prismoid and the formula is called the prismoidal equation. It is easily deduced by simply substituting areas for ordinates in Simpson’s rule. The prismoid differs from the prism in that its parallel ends are not necessarily equal in area; the sides are generated by straight lines from the edges of the end areas. The prismoidal equation is correct when the figure is a true prismoid. In practice it is applied by taking three successive cross-sections. If the mid-section is different from that of a true prismoid, then errors will arise. Thus, in practice, sections should be chosen in order to avoid this fault. Generally, the engineer elects to observe cross-sections at regular intervals assuming compensating errors over a long route distance.