What is Volume?

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

What is the Definition of Volume?

Volume is most commonly described as capacity. It is that particular quantity that describes the capacity of an object in free space. In other words, the volume of an object is the occupied space by an object in 3-dimensions.

Is volume the same for all 3D Objects?

Volume is not a fixed quantity. Different objects are bound to have two different capacities. As a result, the space that they will occupy in the 3D will also be different. Hence, their volumes will be different as well. If we state an example, the amount of liquid that can be kept in a bottle and a jar of different sizes will be different. Hence, their volumes will also be different.

The most common types of shapes that are seen in the 3D includes cones, cylinders, cuboids, cube, and so on. All of these shapes have proper capacities, and they can be expressed in all 3-dimensions. The volume of each of these shapes in free spaces can be found out.

How is Volume Calculated?

As stated before, the volume of an object can be expressed as the capacity that exists in 3-dimensions. Hence, the calculation of the volume of an object is fairly simple. There are three coordinates for all objects lying in 3-dimensions. If the value of each of the coordinates is known, then the value of an object can be easily found out. If a physical object has a distinct length, breadth, and comes with a proper height or width, then the product of all of these quantities will produce the total volume of the object.

What is the Unit of Volume?

In general, the volume of a physical object is denoted by cubic units. This is the measure of the volume of any object. It can change as per the metric system. For instance, the volume of a physical object will be calculated in cubic metres, if all the three dimensions are measured in meters as well. According to the International System of Units or SI, cubic meters is the standard volume unit. Apart from cubic meters, the usage of cubic inches, cubic foot, and cubic centimeters are also well-known.

What is the Volume of a Liquid?

Like every other physical object, liquids also come with their volumes. Unlike solids, the volume of a liquid is measured differently. If a liquid is poured in a container that has the standard units of measurements, then the volume of the container that the liquid occupies once it is poured in the container is the same as the volume of the liquid.

The measurement of the volume of a liquid is also somewhat different from other solid physical objects. For instance, the liquid volume is expressed in litres. This is the standard measurement for the volume of a liquid.

As for the standard relation, 1 litre is equivalent to exactly 1000 cubic cm.

Since 1 cubic metre is equal to 1000000 cubic centimetres, 1000 cubic cm is equal to 0.001 cubic m.

As a result, we can write that 1 cubic m is equivalent to 1000 litres.

However, this quantity may be too big to measure the volumes that are very little in amount. For such cases, the unit millilitres is used.

1 millilitre is equal to 0.001 litre, which in turn is equal to 1 cubic cm.

Another unit by which the volume of any liquid can be measured is known as Gallons. It is another unit that is mainly common in the North American continent.

By conversion, it is known that 1 liter is equivalent to around 0.264172 US liquid gallon. This is an important unit of measurement, and large water bodies or oil barrels are measured with the help of gallons.

Formula

The formula for volume varies with the shape of the object. However, the concept of finding out the volume remains the same. In all cases, the volume of a physical object is calculated by taking the product of the length, breadth, and height (or depth) of the object. In other words, if the area of an object is multiplied by its height or depth, then also the volume of the object can be calculated.

In mathematical terms, we write-

$v=l\times b\times h$

Here, v stands for volume, l stands for length, b stands for breadth or width, and h stands for height (or depth, where depth is denoted by d).

When we consider the usual physical shapes such as cuboids or cube, finding out the volume is not at all problematic if all the dimensions are known. However, other complex shapes exist as well. These include shapes such as spheres, cylinders, or cones. When the volume of such shapes needs to be calculated, extra parameters, such as diameter, radius, or curved surface dimensions need to be considered as well.

The formula for all kinds of shapes are given below-

What is the Volume of a Cube?

The volume of a cube is given as –

$v=a^3$

Here, v is the volume, and ‘a’ is the length of each side of the cube. Since a cube has all sides of equal length, the volume equals the cube of the length of any one edge or side.

What is the Volume of a Cuboid?

The volume of a cuboid is given as-

$v=l\times b\times h$

If the volume for a rectangular prism is required, then it is calculated with this formula, as a rectangular prism is also a cuboid.

This is nothing but the general formula of finding the volume. Since a cuboid has distinct length, breadth, and height, this formula holds for a cuboid.

What is the Volume of a Cone?

The volume of a cone is given as-

$v=\frac{1}{3}\pi {\text{r}}^2\text{h}$

Here, the height of the cone has been denoted by ‘h’, and the radius of the cone has been denoted by ‘r’.

What is the Volume of a Cylinder?

The volume of a cylinder is given by-

$v=\pi {\text{r}}^2\text{h}$

Here, ‘r’ stands for the radius of the cylinder, and ‘h’ stands for the cylinder’s height.

It is to note that when a cylinder and cone comes with the same height and radius, then the cylinder’s volume is three times the cone’s volume.

What is the Volume of a Sphere?

The volume of a sphere is given by-

$v=\frac{4}{3}\pi {\text{r}}^3$

Here, ‘r’ denotes the sphere’s radius.

Practice Problem

If a cuboid has length = 5 cm, breadth = 3 cm, and volume = 60 cubic cm, then find the height of the cuboid.

⇒ Here, l = 5, b = 3, and v = 60

We know,

$\begin{array}{l}v=l\times b\times h\\ =>60=5\times 3\times h\\ =>h=\frac{60}{5\times 3}\\ =>h=\frac{60}{15}\\ =>h=4\end{array}$

Therefore, the height of the cuboid is 4 cm, which is the required answer.

Context and Applications

Volumes are an integral part of the syllabus in mathematics. From school exams to college exams, to exams such as NEET, IITJEE, CMI, and ISI, the concept of volumes is required everywhere.

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

• Bachelor of Science in Mathematics
• Master of Science in Mathematics