Make a mold of the object and then fill the mold with water and measure the volume of the water.
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If the object is irregular the best way is to measure the volume of water it displaces when you immerse it completely in water.
I don't believe there is a way to find the mass of an object knowing only the diameter of the object. If you had the volume, or some other measurements sure. the best bet would be just to weigh it, or find the volume using the principles of displacement.
The answer depends on the object. If the object is of a regular shape the best option would be to measure its dimensions and use geometry to calculate the volume. If the object is small you may be able to find its volume by fluid displacement using a graduated flask (or similar). You will have to find a fluid that does not interact with the object and which is less dense than the object. Fill a graduated flask with the fluid and record the volume. Gently immerse the object and read off the new volume. The difference between the two volume readings will be the volume of the object. Obviously, you will not be able to use water to find the volume of a sugar cube or salt crystals, nor of a block of most kinds of wood. For moderately large objects, you can use two large containers: for example, a bucket and a big tray. Stand the bucket in the tray and fill the bucket to the top with the inert fluid. Then very gently introduce the object into the bucket, collecting the overflowing fluid in the tray. Measure the amount of fluid in the tray using a measuring cylinder or similar. To improve your accuracy, you must lower the object very gently. You can also use additives to reduce the surface tension of the fluid (liquid soap for water). Finally, if the object is big: for example as big as a house, I give up! Challenge your teacher!
Yes. Volume is measured in cubic units.
Simply: Volume (V) is how much space an object takes up whereas density is how "compressed" an objects mass (m) is. Density = m/V Example: Imagine a ton of feathers and a ton of iron. They both have the same mass (weight) but they have different volumes and density.