Volume from Geometry, and the Order That Follows

Three shapes account for nearly all of it

Most volumes on this site are a prism: a plan area carried down through a constant depth. A slab, a screed, a bed of mulch, a layer of drainage stone, a plaster coat — all the same arithmetic, and all of them the product of a shape somebody measured and a thickness somebody chose.

The second shape is the cylinder, for anything bored or cast in a tube: a pier, a post hole, a sonotube column. The third is the frustum — a prism whose top and bottom differ — and it exists because real excavations are not prisms. Everything below about batter and bulking is a consequence of that third shape being used where the first was assumed.

The reason to say this plainly is that the arithmetic is the easy half. Nobody gets the multiplication wrong. What goes wrong is the depth, the sides, and what the volume is then turned into.

The depth carries the risk, and it is not measured

A plan area is measured and is usually right: a slab is a rectangle, a bed is a shape somebody paced out, and the dispute over it is small. The depth is DECIDED, and the decision is frequently made by somebody who is not holding the tape.

Because volume is linear in depth, a proportional error in depth is a proportional error in the order — and depths on site are small numbers where small absolute errors are large relative ones. Twenty-five millimetres extra on a hundred-millimetre slab is a quarter more concrete. The same twenty-five millimetres on a plan dimension of five metres is half a percent.

The second trap is which depth is meant. A granular layer specified as a compacted thickness needs more material delivered than that thickness suggests, because it arrives loose and is then compacted into place — the two volumes differ by a bulking factor that belongs to the material, not to the job. Every page here states which of the two it wants; where it wants the compacted one, the loose volume to order is larger.

One volume, three different orders

A volume is not an order until somebody says how the material is sold, and the three routes round differently.

SOLD BY VOLUME — ready-mixed concrete, screed, bulk aggregate. The supplier delivers in increments, so the order rounds up to the next increment rather than to the next unit, and a job needing a fraction over a full load pays for the load. This is the one case where ordering slightly MORE is usually cheaper per unit than ordering exactly.

SOLD IN BAGS. The count is a ceiling function, and the divisor is the bag's YIELD rather than its weight — see the next section. Part bags do not keep.

SOLD BY MASS — aggregate, sand, riprap, topsoil by the tonne. The volume is multiplied by a bulk density, which is a property of the material as delivered, including its moisture. A wet load weighs more and covers the same ground, so buying by mass in the rain costs more for the same result.

Where it fails: a bag's yield is not its weight

A bag of concrete or mortar mix does not produce a volume equal to its weight divided by the density of concrete. It produces less, and the reason is that the mix is GRADED: the sand occupies the voids between the coarse aggregate, and the cement paste occupies the voids between the sand. The finished material is denser than the loose contents of the bag, so the bag makes less of it than a naive division suggests.

The consequence is that a yield figure cannot be derived and has to be read. Manufacturers publish it per bag, and it differs between products of the same weight because the grading differs. Deriving it instead — by dividing the bag's weight by the density of hardened concrete — overstates the yield and under-orders the job, which is the direction that stops a pour.

The same effect runs the other way for anything bought loose and then compacted, and it is the reason the site keeps bulk densities in a reference table rather than inside each calculator: one material, one figure, cited once.

Where it fails: the sides are not vertical

The prism formula assumes vertical faces. An excavation has none. Trench sides are battered back for stability, a pit needs working room beyond the structure it is being dug for, and a post hole is often belled at the base. Every one of those holds MORE than the prism says, and the discrepancy is not small.

A worked case, because the magnitude is the point. A trench 600 mm wide at the base and 1 m deep, with both sides battered back at forty-five degrees, has a top width of 2.6 m and a cross-section of 1.6 m² — against 0.6 m² if the sides were vertical. That is **2.7 times** the spoil to remove and 2.7 times the backfill to bring, on a shape that looks on the drawing like a rectangle.

The same error appears with the opposite sign on the backfill: the structure in the trench displaces some of it, so the volume to bring back is the excavation less the structure, not the excavation. A page that computes one and not the other is answering half the question, and every page here says which half it is answering.

The alternative: average end areas, and when a prism will not do

For anything whose cross-section changes along its length — a graded road, a swale, a trench that deepens to a fall — the standard method is AVERAGE END AREAS: take the cross-section at intervals, average each adjacent pair, multiply by the distance between them, and sum. It is the method every earthworks package uses and it is what a prism is a single-interval special case of.

It is not exact either. Averaging two end areas assumes the section varies linearly between them, which over-estimates a section that bulges and under-estimates one that pinches; the prismoidal formula corrects for that by taking the middle section as well. Over the interval lengths used on site the difference is usually small, and where it is not, the interval is too long.

This site does not implement average end areas, and the reason is honest rather than technical: it needs a survey — a set of sections at known chainages — and a calculator taking one depth cannot have one. Where the ground varies, the answer here is a starting figure and the section survey is the real one.

Some shapes defeat a single prism without any survey, because they are several prisms at once — a raft and a flight of steps. A raft's plan gives the area and its section gives every thickening under it — the edge band round the perimeter, whose centre line is the perimeter less four times its width so that the corners count once, the downstands and the pads. A flight of solid steps is a stack of prisms each running down to the ground, so n steps hold width × riser × going × n(n + 1)/2; a waist-slab flight is its sloping slab plus a triangle for each step.