How can you calculate the volume based on the liquid level in a spherical tank?

Answer

Suppose that in a spherical tank of radius R the level is at height h. Then you have volume

V = pi (Rh² – h³/3)

(we call that the ‘fundamental formula’)
it is the result of the function’s volume-revolution integral

y = square root [  R² – ( x – R )² ]

(a circle with radius R and center (R,0))
after rotation around the x-axis of the section x = 0..h

For h = 0 this becomes 0,
for h = 2R ( = tank completely full) this becomes 4/3 pi R³ the volume of a sphere

—–

If your tank is cylindrical with two hemispheres, it is best to divide the volume. Suppose the two hemispheres have radius R, and the cylinder itself has height H and radius R.
Suppose the liquid is at height h:

* As long as h < R you can use the basic formula above for V, because then you are still filling up the bottom sphere. h will therefore lie between 0 and R * If h > R but smaller than R+H you are in the cylinder part. In total you then have a volume
Vol = 2/3 pi R³ + pi.R².(hr)
the first piece is the fully filled hemisphere, the 2nd piece the part of the cylinder that is filled.

* finally if h > R+H but smaller than R+H+R you are filling the top hemisphere. The completely filled cylinder then has a volume of pi.R².H. For the part that is in both hemispheres together you first do h = h – H and then apply the basic formula with that value.
So total:
Vol = pi.R².H + basic formula with value h which will then lie between R and 2R