Area of part of a sphere?

I have a crater-like area on my property that was at one point an in-ground pool installed by the previous owner. He then removed the pool and filled it in, but over the years the fill must have settled, and I’m left with a circular crater, about 15’ in diameter. At its lowest point, it’s about 2’ below grade.

I’d like to fill it in and am puzzling over how to determine the volume of the crater, so I know how much fill to have delivered. I can find formulas for determining the volume of a sphere and hemisphere, but not this- I’m not even sure if this shape has a name.

If it were a circle I’d be talking about the chord BX(, but as a sphere I don’t know what it is, or how to determine the volume.

Any ideas?

How about the volume of a spherical cap?

(scroll down)

By garlic, I think that’s it.

Four whole minutes- pretty amazing.


You’re welcome.

Something else to keep in mind: the fill dirt which they bring you is typically measured loose, in the truck (LVM). The amount that it will take to properly fill the void would be measured “in-place”. There will be a difference in the volume.

For example, 3 cubic yards LVM might only pack into 2 cubic yards in-place. For roadway fill, the ratio is about 1.5. For your yard, you certainly won’t be compacting it that much, but you should allow for settlement etc. I’d use maybe 1.2.

IOW if you calculate that you will need 2 cy in-place, order 2.4 cy LVM. Or more, if you have a place to stockpile it … over time, the area will settle and you can just add more from your pile.

That’s a good point. I tend to purposely over-order in situations like this because I figure I can always use it elsewhere if there’s too much. This fill project is just one of several I’ll be starting over the next couple of years, so any excess will just get moved to the next project.