Excavation and muck-away
Excavation with battered sides (prismoidal formula)
- where
- top length (m)
- base length (m)
- batter, horizontal per 1 vertical
- depth (m)
- top width (m)
- base width (m)
- plan area at mid-depth (m²)
- excavated volume (m³)
Explain this equation
This works out the volume of a pit dug with sloping (battered) sides, such as a basement, tank or soakaway dig, and how much more that is than a vertical-sided hole.
L and W are the base dimensions and D the depth. k is the batter: how far the side steps back horizontally for every metre down. Lt and Wt are the top dimensions, and Am is the plan area halfway down.
The prismoidal rule takes the bottom area, four times the mid-depth area and the top area, adds them and multiplies by D/6. For a pit with flat sloping faces this gives the exact volume, unlike averaging top and bottom.
Sense check: a 2 × 2 m base, 1 m deep with a 45° batter (k = 1) opens to 4 × 4 m at the top and holds about 9.3 m³, more than double the 4 m³ of a vertical hole.
Watch out: the safe batter depends on soil and water and must be set by a competent person.
Volume of a pit or basement dig whose sides are battered back at a constant slope, from the base size, depth and batter.
Enter your values to see the result.
Good to know
The batter k is the horizontal distance per 1 m of depth: k = 1 is a 45° slope, k = 0.5 about 63°, k = 2 about 27°. The prismoidal formula is exact for a pit with flat sloping sides, so no averaging error. A safe batter angle depends on the ground and water; it must be decided by a competent person (HSE: the slope should be flatter than the natural angle of repose, much flatter in wet ground).
Common mistake
Using the average of top and bottom areas times depth. That overestimates a battered pit; the prismoidal formula gives the true figure.
Limits
Same batter on all four sides, level base and level ground. Stepped (benched) sides: add each step as its own rectangular dig.
Sources
- Prismoidal formula V = h/6·(A1 + 4Am + A2), exact for a pit with planar battered sides (standard mensuration, e.g. Uren & Price, Surveying for Engineers)
- HSE, Excavations (support, battering, spoil, edge protection, inspection; a cubic metre of soil can weigh over 1.5 tonnes)
Results are calculations from the published methods and figures shown, and depend on the data you enter. Follow BS 8500, BS EN 206, BS EN 752, Building Regulations Part H, manufacturers’ data and your engineer’s design. Structural concrete and foundations need a competent designer.