I have no idea what you mean here. Exotic containers? I'm talking about something like a measuring cylinder or a straight-sided glass or mug.

This StackOverflow answer probably does the subject more justice than I can: https://earthscience.stackexchange.com/questions/14587/what-...

but I'll try and explain it succinctly:

What you're measuring is the volume of rain falling per unit area in a given time (usually 24 hours). If you're collecting rain in a vessel, you divide the volume collected by the area of the vessel's mouth to get the volume collected per unit area. And in order to measure the volume in the container, you measure the height of the water and multiply by the area of the vessel's bottom.

If the bottom and mouth have the same area, those cancel out and you can just specify the rain height regardless of the size of the container. That is, if you have 1ml of rain falling per square mm over 24 hours, it will produce the same height of water if you set out a big container or a small container, as long as the containers have straight walls.

If they don't have the same area, then you can't use the handy mm unit.

I couldn't have explained it better myself. Glad you understand!

Aside from the last bit. The rainfall per unit area is not relevant to the shape of a particular container. 5mm of rain falls regardless of your container shape. Whether that 5mm of rain falling, also means that the water in your container is 5mm deep, is a function of your container shape.

It's not a unit problem. If you're trying to measure rainfall in a conical vessel, you can do that, and the conversion from collected volume to fallen rain will still yield the same 5mm out of the sky.