# Equation for calculating the volume of liquid in an ellipsoid

I am looking for an equation to calculate the volume of liquid in an underground tank based on a depth reading.

The tank is an ellipsoid shape with the following dimensions:

X-Axis Radius:    30.425 cm
Y-Axis Radius:    30.425 cm
Z-Axis Radius:    60.95 cm


I am trying to find an equation that would provide the volume of liquid in liters at a specific depth: For example:

When the tank is filled up with liquid to:
30cm (1/2 way)
the volume of liquid would be around:
120 Liters.

And when the tank if filled up to the top:
60cm
the volume would be around:
240 Liters.


Thank you for your assistance!

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Let the semi-axes (the 'radii') in each direction be $r_x$, $r_y$, and $r_z$, and let the depth of the liquid be $d$. Squeeze (or stretch) the ellipsoid into a sphere of unit radius, so that all of these are $1$; the depth of the liquid is then $d' = d/r_z$, and its volume is $\pi d'^2(3-d')/3$, as you know from your previous question. Now stretch (or squeeze) the sphere back into its original shape, thus expanding the volume of the liquid by a factor $r_xr_yr_z$.