Expert: Steve Holleran - 11/21/2007

Question
I have a volume related question for excavation.  If I am burying a large concrete box-shaped structure, OSHA requires that the hole be dug at a 1 to 1 slope up and away from each of the sides of the core excavation. This gives us a square bottom, but from each side of the square the hole slopes up at a slope of one. We are looking to find the volume of dirt excavated in order to dig such a hole.  The core and sides are an easy calculation, but at each corner, where the two slopes meet, an odd slant-pyramid shape must be dug out.  This is because two sloped edges meet at a single sloped line directly out and up from each corner of the rectangular base to the surface.  Is there a formula for such a shape?

Answer
Hi Clayton,

Okay, I think I have this under control.  Basically, what you're describing is called a frustum of a pyramid.  The bottom square is the lower base, the square that manifests itself on the surface of the ground is the upper base.  What you need to know are the following:

The vertical (perpendicular) height of the hole , h.
The area of the lower base, B1
The area of the upper base, B2.

Then, the volume of the frustum (the hole you're describing) is given by:

        V = (h/3) * (B1 + B2 + sqrt(B1 * B2))

I hope this is what you were looking for, and that it helps you out.
Steve