Volume

Introduction to Volume

**Volumes of a Cube and a Cuboid**

Abhinav’s mother gives him a container, asking him to go to the neighbouring milk booth and buy 2.5 L of milk. What does ‘2.5 L’represent? It represents the amount of milk that Abhinav needs to buy. In other words, it is the volume of milk that is to be bought.

After buying the milk, Abhinav notices that the container is full up to its brim. He says to himself, ‘This container has no capacity to hold any more milk.’ What does the word ‘capacity’ indicate? **The space occupied by a substance is called its volume.** **The capacity of a container is the volume of a substance that can fill the container completely. **In this case, the volume and the capacity of the container are the same. The standard units which are used to measure the volume are **cm**^{3} **(cubic centimetre) **and **m**^{3 }**(cubic metre).**

In this lesson, we will learn the formulae for the volumes or capacities of cubic and cuboidal objects. We will also solve examples using these formulae.

A cube is one among the five platonic solids. This means that it is a regular and convex polyhedron with the same number of faces meeting at each vertex.

**Formulae for the Volumes of a Cube and a Cuboid**

Consider a cube with an edge *a*.

The formula for the volume of this cube is given as follows:

**Volume of the cube = ****a**^{3}

Now, consider a cuboid with length *l*, breadth *b* and height *h*.

The formula for the volume of this cuboid is given as follows:

**Volume of the cuboid = ****l × b × h**

The units of capacity and volume are interrelated as follows:

- 1 cm
^{3}= 1 mL - 1000 cm
^{3}= 1 L - 1 m
^{3}= 1 kL = 1000 L

- A cube has the maximum volume among all cuboids with equal surface area.
- A cube has the minimum surface area among all cuboids with equal volume.

Easy

**Example 1: **

**Find the volumes of cubes of given sides.**

**(a) 2 cm (b) 5 m (c) 12 cm (d) 15 m**

**Solution:**

**(a) **

Measure of side of cube = 2 cm

Volume of cube = (Side)^{3} = 2^{3} cm^{3 }= 8 cm^{3}

**(b) **

Measure of side of cube = 5 m

Volume of cube = (Side)^{3} = 5^{3} m^{3 }= 125 m^{3} ** **

**(c) **

Measure of side of cube = 12 cm

Volume of cube = (Side)^{3} = 12^{3} cm^{3 }= 1728 cm^{3}

**(d) **

Measure of side of cube = 15 m

Volume of cube = (Side)^{3} = 15^{3} m^{3 }= 3375 m^{3} ** **

**Example 2: **

**Find the volumes of cuboids of given dimensions.**

**(a) length = 5 cm, breadth = 2 cm, height = 6 cm **

**(b) length = 15 cm, breadth = 10 cm, height = 30 cm**

**(c) length = 1 m, breadth = 0.5 m, height = 1.5 m**

**Solution:**

**(a) **

We have

l…

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