Suppose there is a spherical tank partially filled with a liquid so that the liquid has a maximum depth of h. I want to find the equation for the volume of liquid in a tank in terms of depth h (and R for radius). I am thinking I should use a triple integral in spherical coordinates but I don't know what to make the bounds since the radius would vary with respect to theta and I want to keep the equation in a general form. Any advice would be appreciated, thanks
2026-04-01 18:52:15.1775069535
Finding the volume of liquid in a spherical tank
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This appears to be known formula, https://mathworld.wolfram.com/SphericalCap.html
Mathworld also provides a step by step derivation for the spherical segment using integration.