I would place it in a container full of liquid after measuring the liquid.
I would measure the amount of liquid displaced by the object.
That amount is equal to the volume of the irregular object if it is fully submerged in the liquid.
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If possible, fully immerse the object in water, and measure the volume of displaced water.
The volumes of small irregular solids are usually easily measured by measuring the volume of fluid displaced when they are immersed in a liquid. For example: You would have to put the object in a measuring glass with water in it. (~1/3 filled) Put the object in it and then get the number from the measuring glass (beaker etc. mostly beaker) and that is how you measure a irregular solid!
Put water in the cylinder to a level that would cover the object. Drop the item in and if it sinks, read the new level on the graduations. The difference in volume between the two marks, is the volume of the item.
Immerse the object in water and measure the volume of water that is displaced. One way would be to fill a container, large enough to hold the object, with water until it is just about to overflow. The container and water would need to be inside another container that could capture the displaced water. Submerge the object in the water and then measure the volume (or weight) of the water that overflows
Because, from Archimedes' Principle, the volume of the water displaced is equal to the volume of the object displacing it. So, were you to collect all the displaced water and put it into a graduated cylinder, it would tell you the volume of the object in question.