# How To Calculate Volume

Atoms bond with other atoms to build molecules. The way they bond determines, among other things, the physical structure of molecules. This means that, depending on the specific atoms involved and the type of bonds they form with each other, molecules can have different geometries and sizes.

When molecules come together to form a macroscopic object they occupy space. The total space they take up depends on their individual size and on possible interactions between the molecules that may affect how they aggregate together. This, in turn, depends on the specific atoms that form the substance.

The consequence is that, depending on the specific chemical species that form a substance, the total space occupied by a certain amount of it will be different. The tridimensional space used up by the substance is called its volume and, as explained before, it depends on the specific atoms in it for a given amount.

Volume is a basic property of matter which is important for different types of applications in mechanics, hydraulics, thermodynamics, chemistry, and other areas. Let’s discover in detail where it comes from and how to calculate it for any given body.

## How to calculate volume

To calculate the volume of a body with common shape or of a body which can be divided into several common shapes, follow these steps:

1. Determine which common shape or shapes make up the body.
2. Calculate the volume of each individual common shape using the following equations:
1. If you are calculating the volume of a body composed of several common shapes, simply add the results together.

## What is volume

Atoms bond with each other to form molecules through different types of bonds. The main types are covalent, ionic and metallic. Next to these, there are also physical  interactions which form weak or temporary bonds between atoms. All of these factors influence how molecules aggregate and how tightly packed they are.

Different types of bonds make atoms sit closer together or further apart within a molecule. This determines the size of the molecule. On the other hand, physical interactions between the atoms in a chemical compound influence the molecule’s geometry. The result (size and geometry) is almost unique to every type of molecule because it depends on the specific atoms contained in it.

The molecules that make up any macroscopic substance, whether in solid, liquid or gaseous state, interact with each other depending on these characteristics. The way they do so determines the space they occupy.

For example, two equal molecules that repel each other on a specific side will only aggregate through a different side. This determines how they will position with respect to each other when thousands of them come together. In turn, this causes a macroscopic sample to occupy a certain amount of space that depends on the specific atoms contained in it, and on their interactions at the nanoscopic level.

Look at the following image. Let’s suppose these two molecules repel each other when approaching as shown above, but they have no problem aggregating as shown below. When millions of them come together, they leave a certain amount of empty space between them, which depends on the way they are positioned with respect to each other. This, in turn, influences the total space occupied by a macroscopic sample made of these molecules. If for some reason they could aggregate as shown above, a sample made up of exactly the same number of molecules as before would have a different size, since the total empty space between them would be different.

The amount of space taken up by any body is called its volume. As mentioned before, this property depends on internal factors, like the chemical species that compose each substance, but it also depends on external factors, like temperature and pressure.

For example, water molecules in gaseous state possess so much thermal energy that they constantly collide with each other and bounce at high speeds. This allows them to fill up any empty space they find. Because of these energetic interactions, molecules in water vapor have an average separation greater than the one they would have at lower temperatures.

Meanwhile, water molecules in ice are closer to each other than in gaseous state. They form a stable framework which exists thanks to weak electrostatic interactions called hydrogen bonds. In this case, if we take a set amount of vapor and freeze it, its volume would decrease due to the smaller distance between each molecule in the second case. In consequence, the total space occupied by the same mass of ice and vapor is different.

Most substances in solid and liquid state have fairly stable volumes. This means the distance between their molecules can’t be easily altered. Think about a metal cylinder. If you stretch it with enough force you will probably increase its length. Nevertheless, it would also become thinner. So, the total space used by the metal cylinder, its volume, would remain constant. Something similar happens with liquids. Their ability to take the shape of the container they are in while preserving their volume is why they are used, for example, in hydraulic systems.

This is not the case with gases, though. If you fill up a balloon with air and you sink it deeper and deeper into the ocean, the pressure acting around it would make it smaller as you dive in. This is because air molecules, as those of many gases, keep greater distances between each other than in liquid or solid state. Therefore, they can be brought closer together without causing them to repel one another. Volume is therefore a very interesting property of matter that reveals its interaction with factors like temperature and pressure.

## How to calculate the volume of common shapes

Common tridimensional shapes, such as spheres, cubes, cones, cylinders, etc., have symmetry axes and features that make it possible to find analytical expressions for their volumes. This means, it is possible to write an equation to calculate the shape’s volume based on its characteristics. This is done through integral calculus, so we will not go into detail about this.

Some of the analytical expressions for common shapes’ volumes are pretty intuitive. For example, to find the volume of a rectangular prism (similar to a cube, but with each side of a different length), we just need to calculate the area of one of the faces and then multiply it by the length perpendicular to said face, as shown in the following image.

The case of a cylinder is similar: it is a shape composed of a circular area that extends over a certain length. In this case, its volume equals the circular area times its height.

What about bodies that do not have a common shape? Take a look at the following gas tank, for example. Although at first glance it is no common shape, you could actually put some common shapes together to produce the same volume.

In this case, the volume of the entire shape can be approximated by the volume of two half spheres at both ends, and of one cylinder in the middle. This way, the total volume would be:

Given that the volume of many bodies can be described as the sum of the volumes of different common shapes “attached” together, it is very useful to have the analytical expressions for their individual volumes at hand. The following image summarizes them.

## How to measure the volume of irregular shapes

How can we determine the volume of a body that can’t be decomposed into several common shapes, for example, an irregular rock or a bronze bust of a historical figure? In this case, the ancient Greeks can give us a hand:

In the 3rd century BC, the famous mathematician and inventor Archimedes allegedly realized that, as he got into a tub to take a bath, the initial water level would rise. This is because water is displaced by his body’s volume. If the initial volume of water was determined prior to him getting into the tub, he could determine his body’s volume by measuring the difference at the end.

This is called Archimedes’ principle, and is a simple method to determine the volume of bodies with irregular shapes, which cannot be practically divided into common shapes. The procedure is very simple:

1. Select a liquid which does not react with the sample whose volume will be determined. Make sure that it also has a lower density so that the body fully sinks in it. Read our article on how to calculate density to learn more about this effect.
2. Use a graduated cylinder to precisely measure the volume of the liquid.
3. Introduce the sample into the liquid inside the graduated cylinder.
4. Measure the final volume.
5. Calculate the volume of the body of interest using the following equation:

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