Question: Hi! I work in the water industry, providing automatic controls for water systems. I was wondering if there was a formula for calculating the volume of water in a sphere shaped water tower? With the use of a pressure gauge we know the elevation of zero water {the bottom of sphere) and the overflow elev. (near the top of sphere) I would like to calculate how many gallons are at any elevation in between.

Thanks Dan


Hi Dan.

Using integral calculus, we can derive an equation for the volume of a sphere between a plane and its surface. That equation is:
V = (pi/3)h2(3r-h)
Where V is the volume, pi is approximately 3.14159, r is the radius of your sphere without respect to the water level and h is the height from the bottom of the sphere to the waterline.

In your question, it sounds as though "zero water" does not equal "zero height" above the spherical bottom, so you should also calculate the zero water volume (using the same formula) and subtract this to get your net water volume.

Once you have the net volume, it will be in cubic length units (meters cubed or feet cubed, whichever you are measuring with). To convert to gallons, you'll need to multiply by the appropriate conversion factor. "Gallons" is somewhat ambiguous here - there are different conversion factors for UK and US gallons, so you'll need to ensure you apply the right conversion factor. In Canada we measure things in litres and meters, so once we have the volume in meters cubed, we just move the decimal over three places.

Hope this helps,
Stephen La Rocque.