Math CentralQuandaries & Queries


Question from William, a student:

Trying to calculate fill dirt for leveling a sloped yard to place a slab. Slab is 30'x40', yard slopes from corner #1 south 40' to Corner #4 drop is 11". From corner #1 east 30' to corner #2, drop is 11 1/4". From corner #1 southeast at a 45 degrees, 50' to corner #3, drop is 19 3/4". Corners numbered in a clockwise direction.


I don't have the shape of the sloped surface so the best I can do is approximate the amount of fill you will need. If the slope were a plane surface, that is not curved at all then the drop from #1 to #3 would be 21 1/4 inches. That is what I am going to assume so my estimate will be a little too large.

My assumption that the base is a plane surface and that the volume is the area of the top times the depth at the center. The area at the top is $30 \times 40 = 1200$ square feet. The center is half way from #1 to #3 so the depth at the center is $\frac12 \times 21 \frac14 = 10 \frac58$ inches. I need this in feet so divide by 12 and then the volume of fill required is

\[\frac{1}{12} \times 10 \frac58 \times 1200 = 1062.5 \mbox{ cubic feet}\]

There are $27$ cubic feet in a cubic yard so this is $\large \frac{1062.5}{27} = 39$ cubic yards.

I hope this helps,

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