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Volume of earthwork batters

I am having some difficulty calculating the volume of an earthworks batter (the sloping shoulder section of an embankment).

I have used the formula for calculating the volume of a pyramid and have also used a method of calculating the average area (in this case the area of one end divided by 2 as the area at the other is 0) and multiplying that by the length.

However the two results I get are vastly different! I have no idea why...

I believe the solution will be straight forward however I cannot figure it out solo and need the wisdom of someone more capable than myself to point me in the right direction.

Please see attached.

(clearly my word processing skills exceed my mathematical ones...)

Re: Volume of earthwork batters

The first volume is the correct one. I guess the question really is why would you expect the volume to be the "mean" area of cross sections time the height to give the correct volume.

(I looked up "mean area method" "volume" on Google and got references to two books, one on surveying and the other on "Railroad field geometry". In both the "mean area method" was an **approximation** to true volume got by using the area of a number of cross sections at different points. The second also referred to the "average end area method" which just used the areas at the two ends and is, at best, a "quick and dirty" estimate.)

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Re: Volume of earthwork batters

Thanks HallsofIvy.

It's been a while since I've done any proper geometry and I *did* actually think the two methods would give the same result (hence my post here after, in fact, it did not). I even consulted two surveyors I work with who both said the End Area Method works! (Not exactly confidence building!)

There's obviously something to do with the third dimension that causes the variation...? I believe the example works if we only look at, say, the base area?

A1 = 1/2bh = 1/2 x 150 x 15 = 1,125mē

A2 = (15 + 0)/2 x 150 = 1,125mē

The reason I wanted to know why there was a difference was because it would be easier for me to develop an excel template to calculate variations to the last example such as the one attached.

Sorry for my juvenile analysis of this problem, but unfortunately this one had me truly stumped.