Volumes and surface areas of construction elements

Concrete, excavation, backfill, aggregates, insulation fill and stockpiles are measured by volume, and formwork, waterproofing and coatings by surface area. This lesson moves from flat shapes to solids. It covers the prism rule that handles most building elements, cylinders and cones for piles, columns and stockpiles, the frustum for tapered elements, the center-line method for strip footings, the contact area that drives formwork costs, and two methods for volumes that change along their length: the average end area method and the more accurate prismoidal formula. Lesson 5 then turns volumes into masses, truckloads and loose or compacted earth quantities.

The key to volume work is consistent units. Convert every dimension to meters, or every dimension to feet, before multiplying. A slab 125 mm thick entered as 125 instead of 0.125 is a thousand times too large, and one 5 in thick entered as 5 instead of 0.4167 ft is twelve times too large. Both errors are easy to make and easy to catch with a rough check.

Prisms: the workhorse rule

A prism is any solid with a constant cross-section along its length. Slabs, walls, beams, strip footings, curbs, trenches of uniform section and columns are all prisms. Their volume is the area of the cross-section times the length.

A rectangular slab is a prism whose cross-section is a rectangle; a wall with a sloping top is a prism whose cross-section is a trapezoid; a curb and gutter is a prism with an irregular cross-section that can be split into rectangles and triangles. Once the cross-section area is known, the length does the rest. For elements that change section, such as a slab with thickened edges, split the element into parts that are each prisms and add their volumes.

Strip footings and the center-line method

A continuous strip footing under the perimeter walls of a building has the same cross-section all the way round, so its volume is its cross-section area times its length. The question is which length. If you use the outside perimeter, the four corners are counted twice; if you use the inside perimeter, the corners are missed. For a rectangular building the correct length is the perimeter along the center line of the footing, which is the outside perimeter less four footing widths.

For buildings with more corners, the rule generalizes: each external corner reduces the center-line length by half a footing width on each of its two sides, and each internal corner increases it by the same amount. For a shape with only right angles, external corners always outnumber internal ones by four, so the simple rule still holds.

Cylinders, piles and columns

Round piles, caissons, sonotube columns and pipe bedding are cylinders: their volume is the area of the circle times the length. For drilled piles, the actual volume of concrete placed is often larger than the theoretical volume because the hole is slightly oversize or the soil sloughs, and specifications usually require the contractor to record the concrete placed in each pile so that it can be compared with the theoretical figure. A pile that takes much less concrete than expected may have a defect, such as a collapse or a void.

Cones, pyramids and frustums

A cone or pyramid holds one third of the volume of the prism or cylinder with the same base and height. Stockpiles of sand and gravel form roughly conical shapes, so the cone formula gives a quick estimate of how much material a pile contains. The angle of the pile's side, called the angle of repose, depends on the material: dry sand and gravel typically stand at about 30 to 40°. If you know the angle and the diameter, the height follows from trigonometry (Lesson 7).

A frustum is a cone or pyramid with its top cut off parallel to the base. Tapered piers, bell-bottomed caissons, hoppers and some footings are frustums. The general formula uses the areas of the top and bottom faces, A₁ and A₂, and the height h.

Surface area and formwork contact area

Formwork is priced by the contact area: the area of form surface that touches the concrete. For a wall it is both faces plus the ends (bulkheads) where the wall stops; for a column it is the perimeter times the height; for an elevated slab it is the soffit plus edge forms; for a beam it is the bottom and both sides. Slabs on grade need only edge forms, measured by length or by the area of the edge. Because formwork labor and materials often cost more than the concrete itself, contact area is one of the most important quantities in a concrete estimate.

Surface areas of solids also matter for coatings and insulation. The curved surface of a cylinder is its circumference times its length, π d L, which gives the area of paint on a round column or the insulation wrap on a pipe or tank. The total surface of a closed tank adds the two ends.

Volumes that change along their length

Trenches with a varying depth, road embankments, ditches and cuttings change cross-section from point to point. Their volume is estimated from cross-sections measured at intervals, usually at regular stations along a center line (Lesson 8). Two methods are used.

• The average end area method treats the volume between two sections as their average area times the distance between them: V = L × (A₁ + A₂) ÷ 2. It is simple and widely used for earthwork payment.
• The prismoidal formula uses the two end areas and the area of a section measured exactly halfway between them, Aₘ: V = (L ÷ 6) × (A₁ + 4Aₘ + A₂). It is exact for prismoids, solids whose sections vary smoothly, and is more accurate when the end areas are very different.

The middle area in the prismoidal formula must be measured or calculated from the middle dimensions, not found by averaging the end areas; using the average of the end areas would simply reproduce the average end area result. For a cone or pyramid, the average end area method overstates the volume by 50 %, which shows why the prismoidal formula is preferred when sections change greatly, for example where a cut runs out to nothing.

Which method is used for payment is a contract matter. Many earthwork contracts and highway specifications name the average end area method because it is simple and repeatable, and accept its small bias. Others allow computer surface models, which compare an original ground surface with a design or final surface and calculate volumes directly; the hand methods remain the way to check those results.

Checking volumes

Volume errors are usually large, so a rough check catches most of them. Before accepting a result, ask three questions. Is the order of magnitude right: does a house basement slab really need 200 m³? Do the units make sense: is the answer in cubic units, and were all dimensions in the same unit? Does it compare well with a simple bounding shape: is the volume of a tapered element between the volumes of the prisms built on its smallest and largest sections? A habit of estimating before calculating, even crudely, is one of the marks of a reliable estimator.