Volume of Cube and Cuboid

We have spoken about surface area but another very important related quantity when it comes to solids and shapes is Volume. The amount of space occupied by a three dimensional object is called its volume. For example: The space is a room is greater than the space in a cabinet. In other words, the volume of the room is greater than the volume of the cabinet. Similarly, the volume of a pencil box will be greater than the volume of the pen and the eraser kept inside it.

In terms of real life applications, how many bottles of water can we place in a small store room? What is the amount of milk that can be carried by the milkman if he has two containers with a fixed value of dimensions? Such problems can only be solved when we check for the related volumes.

Now, how to measure the volume of any of these objects?

We use square units- m2,cm2 etc. for denoting the area of a region. Now, we will be using cubic units to find the volume of a solid, as a cube is the most convenient solid shape (just as square is the most convenient shape to measure area of a region). Just like for finding the area, we divide the region into square units, similarly, to find the volume of a solid we need to divide it into cubical units.

Division of Volume

The volume of a solid can be said to be measured by counting the number of unit cubes it contains. Cubic units which we generally use to measure volume is equal to:

1 cubic cm = 1 cm × 1 cm × 1 cm = 1 cm3 = 10 mm × 10 mm × 10 mm = 1000 mm3

1 cubic m = 1 m × 1 m × 1 m = 1 m3 = 106cm3

1 cubic mm = 1 mm × 1 mm × 1 mm = 1 mm3 = 0.1 cm × 0.1 cm × 0.1 cm = ...................... cm3

Let's start by finding the volume formula for the basic shapes of cube, cuboid and cylinder.

Cuboid

In the below table, consider the sides and assign them the designation of "length" , "breadth" and "height". Now, take the numerical values and find the product of length×breadth×height. (Each cube has a dimension of 1 x 1 x 1 unit)

S.No	Cuboid	l×b×h
(i)		unit3
(ii)		unit3
(iii)		unit3

We observe that the numerical value of the product l×b×h is the same as the number of unit cubes present in the respective arrangement. This product is the volume of the cuboid.

Since, l×b is also the area of the face taken to be the base of the cuboid, we can also write:

Volume of cuboid = Area of base × height

Take 36 cubes of equal size (i.e., length of each cube is same). Arrange them to form a cuboid. What do you observe? Since we have used 36 cubes to form these cuboids, volume of each cuboid is 36 cubic units. Also volume of each cuboid is equal to the product of length, and of the cuboid.

Try These

Find the volume of the following cuboids.

Instructions

Cube Volume

We have already spoken about how, the cube is a special case of a cuboid where l = b = h. By simply putting this condition into the above equation for volume of cuboid:

Instructions

Thus,

Thus,

A company sells biscuits. For packing purpose they are using cuboidal boxes:

(1) Box A = 3 cm × 8 cm × 20 cm

(2) Box B = 4 cm × 12 cm × 10 cm

What size of the box will be economical for the company?

Instructions

Volume and Capacity

Noth the terms "Volume" and "Capacity" are used interchageably. However, there isn't much of a difference between them.

(a) Volume is the amount of space occupied by an object while

(b) Capacity is the quantity of any substance that a container is capable of holding.

For example: If a water tin holds 100 cm³ of water then the capacity of that water tin is 100 cm³.

Note: Capacity is also measured in terms of litres (L).

Example 8: Find the height of a cuboid whose volume is 275 cm3 and base area is 25 cm2.

Instructions

Example 9: A godown is in the form of a cuboid of measures 60 m × 40 m × 30 m. The number of cuboidal boxes that can be stored in it, if the volume of one box is 0.8 m³ is:

Instructions