A common situation in construction estimating and planning is the excavation of a pit or trench with sloped sides. For example:
In this example, we have a trench of unknown length with the sides of the excavation cut back at a slope of 1:1. The bottom of the trench is 5 feet wide. If we assume the depth is also 5 feet, then the top of the trench is 15 feet wide. The cross-sectional area of the trench is (5+15)/2*5, or the average trench width of 10 feet * 5 feet deep = 50 square feet. If the trench is 100 feet long, and the ends of the trench excavation are not also sloped, then the volume is 50 * 100 = 5000 cubic feet, or 185 cubic yards.
Now, what if we have a column footing that is 3 feet x 3 feet and all four sides are sloped? Many would calculate the volume as the average width * the average length * the depth, or 10*10*5 = 500 cf, or 18.5 cy. But that is not the correct calculation. It’s close, but not correct.
If you look at this solid, it looks just like the excavation volume only upside down.
What you are looking at is a pyramidal frustum. It’s a pyramid with the top cut off. The formula for finding its volume is:
The variables are h for the height, A1 for the area of the base, and A2 for the area of the top.
To transform this for use with our excavation, Ab will be the area of the bottom of the excavation, At will be the area of the top of the excavation, and D will be the depth.
So, the formula is:
Ab = Wb * Lb, where Wb and Lb are the width and length of the bottom of the excavation.
At = Wt * Lt, where Wt and Lt are the width and length of the top of the excavation.
In our example, Wb = Lb = 5 and Wt = Lt = 15, so Ab = 5 * 5 = 25 and At = 15 * 15 = 225, and D = 5.
Therefore, the volume is: = 542 cf or 20.0 cy.
This is 1.5 cy more than the 18.5 cy we calculated using the average width and length, or 8% more.
So, it was close but not correct to use the average width for a pyramidal frustum.
The formula can be:
Or, to replace and with the calculated values based on slope, where the slope factor is (1:1 is 1, 0.5:1 is 0.5, etc.):
In this formula, all we need to know are the length and width of the pit bottom, the depth, and the slope factor.
In estimating, approximations are acceptable. We could also argue that the excavation will not actually be done so exactingly and we cannot know the actual excavated quantity. But we have an interest in improving the quality of estimates by eliminating errors that are under our control. Using the average width to calculate the volume of a sloped sided pit is an incorrect calculation of the quantity we want, not an approximation.
Some will say, “But that’s a more complex formula and I won’t remember it.” This is where there is added value in a knowledgebase that already has the formula built in to produce the calculation for you – the same way – every time. You don’t have to remember the formula or perform the calculation. You just provide the dimensions and you get the correct answer – every time.