Volume: Definition, Unit, Examples, and Formulas

Volume is a mathematical quantity that describes the capacity of a three-dimensional object. For example, the amount of milk a cylindrical container can occupy is understood by its volume.

There are different shapes we can observe around us. Different shapes can have different volumes, and we use different formulae to calculate the volume of different shapes such as a cube, cuboid, cylinder, cone, etc. These shapes are defined as three-dimensional objects. For all these shapes, we will discuss the methods of finding the volume.

Definition of Volume

The volume of a three-dimensional solid is the amount of space it occupies or space enclosed by a boundary or occupied by an object or the capacity to hold something. For example, the volume of a cuboidal box indicates the amount of water or any substance contained in it.

In the case of a box full of sand, the amount of sand can be filled inside the box without leaving any gaps in the volume of the box.

Unit of Volume

The volume of solids is measured in cubic units. For example, if the dimensions are given in \(m\), then the volume will be in \({m^3}\). This is the standard unit of volume in the International System of Units (SI). Similarly, other units of volume are \({\rm{c}}{{\rm{m}}^3},\;{\rm{m}}{{\rm{m}}^3},{\rm{inc}}{{\rm{h}}^3}\) etc. To measure the volume of liquids we often use litres.

We know,

\(1\) litre \( = 1000\;{\rm{c}}{{\rm{m}}^3}\)

\(1000\;{\rm{c}}{{\rm{m}}^3} = \frac{{1000}}{{100 \times 100 \times 100}}\;{{\rm{m}}^3} = 0.001\;{{\rm{m}}^3}\)

\(0.001\;{{\rm{m}}^3} = 1\) litre

Hence,

\(1\;{{\rm{m}}^3} = 1000\) litre

Examples of Volume

Let us easily understand this. Try to fill an ice tray with water (Ice tray contains eight cubic moulders). Pour the water into a beaker and measure how much water is in that. Now, if we divide the result by twelve, we will get the volume of each cubic moulder in the ice tray.

The knowledge of volume is essential in calculating the mass of something. Suppose if we need to find the mass of a metal cuboid, we can find out if we know the density of the material as:

Volume Formulas

For cube and cuboid, it is easy to find the volume as they have flat surfaces. But for curved shapes like a cone, cylinder, and sphere, the radius or diameter of its curved surface well

plays a major role to find their volume.

Let us talk about the volume of some three-dimensional shapes:

1. Volume of a cuboid
2. Volume of a cube
3. Volume of a cylinder
4. Volume of a cone
5. Volume of a sphere
6. Volume of other shapes

Volume of a Cuboid

A cuboid is a prism. The volume of a cuboid can be calculated when the length, breadth, and height are given. A cuboid can have a different measure of side lengths and the length of the edges of a cube are the same.

We know the volume of any three-dimensional figure \({\rm{ = area of the base \times height}}\)

A cuboid has six rectangular surfaces. Let us consider it as the base.

Since the area of the base \({\rm{ = length \times breadth}}\)
Volume of the cuboid \(d = {\rm{length \times breadth \times height}}\)

Volume of a Cube

Volume of a cube when the edge or side is given:

A cube is a prism. The volume of a cube can be calculated when the edge or side is given. The method of finding the volume of a cube is similar to calculating the volume of a cuboid. A cuboid can have different measure side lengths, and the length of the edges of a cube are the same.

If we know the length of one of the sides of the cube we can easily find out the volume of it. So, the volume of a cuboid \( = {\rm{length \times breadth \times height}}\)

Volume of a cube \({\rm{ = side \times side \times side = ( side }}{{\rm{)}}^{\rm{3}}}{\rm{[length = breadth = height]}}\)

Volume of a cube when the diagonal is given:

We discussed earlier that a cube has six square faces and all twelve sides of it are equal. Thus, all possible diagonals are also equal.
Let us say the side of a cube is \({\rm{a}}\,{\rm{units}}\) and the diagonal is \({\rm{d}}\,{\rm{units}}\)

Therefore, \({\rm{length = breadth = height = a units}}\)

Since the formula of the diagonal of a three-dimensional object is \( = \sqrt {{{({\rm{ length }})}^2} + {{({\rm{ height }})}^2} + {{({\rm{ breadth }})}^2}} \)

We can derive easily the diagonal of a cube from the above formula by substituting \({\rm{length, breadth, height}}\)  by \({\rm{a units}}\)

Hence, \(d = \sqrt {{{(a)}^2} + {{(a)}^2} + {{(a)}^2}} = \sqrt 3 a\)

\( \Rightarrow d = \sqrt 3 a\)

\( \Rightarrow a = \frac{d}{{\sqrt 3 }}\)

The volume of the cube when the diagonal is given \( = \frac{d}{{\sqrt 3 }} \times \frac{d}{{\sqrt 3 }} \times \frac{d}{{\sqrt 3 }}\)
\(= \frac{{{d^3}}}{{3\sqrt 3 }}\)

The volume of the cube when the diagonal is given \( = \frac{d}{{\sqrt 3 }} \times \frac{d}{{\sqrt 3 }} \times \frac{d}{{\sqrt 3 }}\)
\(= \frac{{{d^3}}}{{3\sqrt 3 }}\)

Volume of a Cylinder

The cylinder is a three-dimensional solid. The volume of three-dimensional shapes is the product of its area of base and height. The volume of a cylinder is equal to the product of the circular base’s area and the cylinder’s height.

Volume of a cylinder \({\rm{ = area of the circular base \times height}}\)

Area of a circle\( = \pi {r^2}\). The height of the right circular cylinder is \(h\)

∴ Volume of a cylinder\( = \pi {r^2}h\)

Volume of a Cone

In general, a cone is like a pyramid with a circular base. We can easily calculate the volume of a cone if its height and radius are known. Here, \(r\) is the radius of the circular base and \(h\) is the height, and \(l\) is the slant height of the cone.

Hence, the volume of a cone \( = \frac{1}{3} \times \pi {r^2} \times h = \frac{1}{3}\pi {r^2}h\)

Volume of a Sphere

(a) Volume of a sphere when the radius of the sphere is given

The volume of a sphere can be calculated using the formula,

The volume of a sphere is \(V = \frac{4}{3}\pi {r^3}\)cubic units
where \(r\) is the radius of the sphere

(b) Volume of a sphere when the diameter of a sphere is given

The volume of a sphere is \(V = \frac{4}{3}\pi {\left( {\frac{d}{2}} \right)^3}\)

where \(d\) is the diameter of the sphere.

Volume of Other Shapes:

Solved Examples

Q. 1: Find the length of the height of the cuboid if its base area is \(120\;{\rm{c}}{{\rm{m}}^2}\)  and the volume is \(1800\;{\rm{c}}{{\rm{m}}^3}\).

Ans:

Given, the volume of the cube\( = 1800\;{\rm{c}}{{\rm{m}}^3}\)

Let the length of the height is \(h\)

We know,

The volume of a cuboid \({\rm{ = length \times breadth \times height = areaofthebase \times height}}\)

Substituting the value, we get,

\(120 \times h = 1800\)

\( \Rightarrow h = \frac{{1800}}{{120}} = 15\)

\( \Rightarrow h = 15\;{\rm{cm}}\)

Thus, the height of the cuboid is \(15\;{\rm{cm}}\)

Q. 2: Find the length of the sides of the cube if its volume is\({\rm{216c}}{{\rm{m}}^3}\).

Ans:

Given, the volume of the cube \({\rm{ = 216c}}{{\rm{m}}^3}\)
Let the length of the sides is \(a\)
The volume of a cube \( = {({\rm{ side }})^3}\)

Substituting the value, we get,

\({a^3} = 216\)

\( \Rightarrow a = \sqrt[3]{{216}}\)

\( \Rightarrow a = 6\;{\rm{cm}}\)

Thus, the side of the cube is \(6\;{\rm{cm}}\)

Q. 3: If the diagonal is \(4\sqrt 3 \;{\rm{cm}}\). Find the volume of the cube.

Ans:

We know,

\(V = \frac{{{d^3}}}{{3\sqrt 3 }}\)

\( \Rightarrow V = \frac{{{{(4\sqrt 3 )}^3}}}{{3\sqrt 3 }} = \frac{{64 \times 3\sqrt 3 }}{{3\sqrt 3 }}\)

\( \Rightarrow V = 64\;{\rm{c}}{{\rm{m}}^3}\)

Hence, the volume of the cube is \(64\;{\rm{c}}{{\rm{m}}^3}\)

Q.4: Find the volume of a cylindrical shape oil container, that has a height of \(8{\rm{cm}}\) and diameter of \(12{\rm{cm}}\) considering  \({\bf{\pi }} = 3.14\)

Ans: Given,

Diameter of the container \( = 12\;{\rm{cm}}\)
Thus, the radius of the container \( = \frac{{12}}{2}\;{\rm{cm}} = 6\;{\rm{cm}}\)
Height of the container \( = 8\;{\rm{cm}}\)
The formula of the volume of a cylinder \( = \pi {r^2}h\)

Therefore, the volume of the given container \(r = \pi {(6)^2} \times 8\;{\rm{c}}{{\rm{m}}^3}\)

Volume\( = 3.14 \times {(6)^2} \times 8\)
\( = 904.32\;{\rm{c}}{{\rm{m}}^3}\)

Hence, the volume of the cylinder is\(904.32\;{\rm{c}}{{\rm{m}}^3}\)

Q.5. Find the volume of a sphere whose radius is  considering\(\pi = \frac{{22}}{7}\)

Ans : Given the radius of a sphere, \(r = 6\;{\rm{cm}}\)

We know that the volume of a sphere is calculated as

\(V = \frac{4}{3}\pi {r^3}\)

So, the volume of a sphere of radius \(6\;{\rm{cm}} = \frac{4}{3}\pi \times {6^3}\;{\rm{c}}{{\rm{m}}^3}\)

\( = \frac{4}{3} \times \frac{{22}}{7} \times 6 \times 6 \times 6\;{\rm{c}}{{\rm{m}}^3}\)
\( = \frac{{4 \times 22 \times 6 \times 6 \times 2}}{{3 \times 7}}\;{\rm{c}}{{\rm{m}}^3}\)
\( = 905.14\;{\rm{c}}{{\rm{m}}^3}\)

Summary

The volume is a mathematical quantity that describes the capacity of a three-dimensional object. We have learned how we use different formulae to find the volume of different shapes.

Check NCERT Solutions for Surface Areas and Volumes

Related Topics

FAQs

Q.1. What are the examples of volume?

Ans : (a) If a cup can hold \(100\,{\rm{ml}}\) of water up to the brim, then its volume is said to be \(100\,{\rm{ml}}\).

(b) If a bottle can hold \({\rm{1 – litre}}\) milk, then its volume is said to be \({\rm{1 – litre}}\).

Q.2. What is volume in simple words?

Ans : The volume describes the capacity of any three-dimensional object. The volume of a cuboidal box indicates the amount of water or any substance that can be contained in it.

Q.3. What is the volume of liquid measured in?

Ans : The volume of liquid can be measured in the \({\rm{litres (l), millilitres (ml)}}{\rm{.}}\)

Q.4. What is volume in maths?

Ans : The volume of a three-dimensional solid is the amount of space it occupies or space enclosed by a boundary or occupied by an object or the capacity to hold something.

Q.5. Are size and volume the same?

Ans : No, the size and the volume are not the same. The volume is a well-defined mathematical quantity that describes the capacity of a three-dimensional shape but the size is not the well-defined term as it can refer to the magnitude of dimensions, area, etc., of any shape.

Q.6. What are the formulas for volume?

Ans : The formula of volume of the cuboid,\(V = {\rm{length \times breadth \times height}}\)

The formula of volume of a cube, \(V = {\mathop{\rm side}\nolimits} \times {\mathop{\rm side}\nolimits} \times {\mathop{\rm side}\nolimits} = {({\rm{ side }})^3}\)

The formula of volume of a cylinder, \(V = \pi {r^2}h\)

The formula of volume of a cone, \(V = \frac{1}{3} \times \pi {r^2} \times h = \frac{1}{3}\pi {r^2}h\)

The formula of volume of a sphere, \(V = \frac{4}{3}\pi {r^3}\)

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