Can you calculate the volume based on the liquid level in a spherical tank and how? Furthermore, how can you calculate the volume of a cylindrical tank whose both ends are hemispheres?

## Answer

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

V = pi (Rh^{2} – h^{3}/3)

(we call that the ‘fundamental formula’)

it is the result of the function’s volume-revolution integral

y = square root [Â R^{2} – ( 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 R^{3} the volume of a sphere

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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 R^{3} + pi.R^{2}.(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.R^{2}.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.R^{2}.H + basic formula with value h which will then lie between R and 2R

## Answered by

#### prof.dr. Paul Hellings

Department of Mathematics, Fac. IIW, KU Leuven

Old Market 13 3000 Leuven

https://www.kuleuven.be/

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