Depending on the size of the object this can be difficult. For a large object, lime a mountain, say, you have no option but to estimate.
For smaller objects use a large beaker, and fill it with a liquid in which your shape is not soluble. Measure the volume of liquid in the beaker. Then place the object into the water and make sure it is totally submerged. The volume of the water in the beaker will rise and the volume of the shape is the difference between the two measures.
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It is the displacement method.
First 'weigh" the ojbect on a scale. this gives the mass. 2nd use the water displacement method to find the volume. Then divide Density = mass / volume.
Volume displacement is the method used to find the volume of small or irregularly shaped objects by noting the difference in the level of liquid before and after after immersing an object into a graduated cylinder or beaker of liquid. The difference between the before and after levels of the liquid is the volume of an immersed object.
If it is a small shape that is denser than a convenient fluid and insoluble in it and also does not react with it then the displacement method is simplest. Fill a graduated cylinder with the fluid, measure the volume of the fluid. Then gently insert the shape and measure the apparent volume of the fluid. The difference between the two volume readings is the volume of the shape. If the shape is less dense than the fluid you have to make one change. You need to find a dense insoluble object. Measure the volume of the fluid with the dense object immersed in it. Then measure the volume when the dense object and the shape are joined together and submerged. The difference between the two measures is the volume of the shape. This method will not work with soluble shapes unless you can find a fluid that it is not soluble in. Similarly, you cannot use a fluid that will react. So measuring the volume of a lump of sugar or a lump of sodium using water are non-starters. Finally, the method will not work if the irregular shape is huge.
Half fill a graduated glass beaker with water and note the level. Place irregularly shaped object into the beaker. Note the new level. The difference between the two levels is the volume of the object. cw: OK, that method works for objects that don't get wet -- like steel. Tissue paper would not be a good candidate. Also, if the object floats, you have to submerge the object.