If you have an object with a standard geometric shape, such as a cube or a sphere, you can calculate its volume by measuring its dimensions and using the relevant mathematical formula. For example, if you measure the length of one side of a cube (L), the volume of the cube is L3. The volume of a sphere would be V = (4÷3)πr3. What do you do with an irregular object, though, such as a pen or a rock? The Greek philosopher Archimedes was faced with this problem when he was asked to find the density of the king's crown. To determine its density, he had to know its volume, and his "Eureka" moment occurred when he realized he could do it by immersing the crown in water and measuring the amount of water that was displaced. The displacement method is still the standard way to determine the volume of an irregularly shaped object.
You can find the volume of an irregularly shaped object by measuring the amount of water it displaces. If you know the density of the object, you can also find its volume simply by weighing it.
Using the Displacement Method
Find a Suitable Container
Look for a container large enough to hold the object. It's best to use a container with a regular shape, such as a cylinder or a box, because you'll need to calculate its volume. If you don't have a regular container, you can always fill it to the brim with water, catch the water that overflows when you immerse the object you're measuring and transfer the water to a graduated vessel.
Fill the Container with Water
Add at least enough water to allow you to completely immerse the object. If you aren't using a graduated container, mark the water level on the side of the container.
Immerse the Object and Note the Change in Water Level
Read the new level on the scale if you use a graduated vessel. Subtract the old level from this one to get the change in level.
If you don't use a graduated container, make a new mark on the container. Subtract the height of the original mark from the height of the new one to get the change in water level.
Calculate the Volume of Water Displacement
Read the volume levels if you use a graduated container, but if you use an ungraduated container, you'll have to calculate the volume. The calculation depends on the shape of the container.
Cylindrical Container: Measure the radius of the container opening (r) and calculate volume of displaced water using this formula: Volume = πr2 • (change in water depth).
Rectangular Container: Measure the length (L) and width (W) of the container opening. The volume of displaced water is: L • W • (change in water depth).
It's Easier When You Know the Object's Density
You may be measuring the volume of a copper penny or a silver statue. Both of these have known densities that you can look up. If you know the density of the material from which the object is made, you can find the volume simply by weighing the object. Since density = mass ÷ volume; volume = mass ÷ density.
Example: A silver statue weighs 10 kilograms. Since the density of silver is 10,490 kg/m3, the volume is 10 kg ÷ 10,490 kg/m3 = 0.00095 cubic meters. 1 cubic meter equals 1,000 liters, so this equals 0.95 liters or 0.25 U.S. gallons.