Volume

The space occupied by any object is called the volume. Volume is three dimensional. To measure the volumes we need to know the measure of 3 sides. Since, volume involve involves 3 sides, so it is measured in cubic units.

Surface like the page of a book, blackboard, etc. are called plane surfaces. They do not have any volume but have only area.

A cube is a solid box whose each surface is a square of same area.

Therefore, cube has: 6 surfaces or faces, 8 vertices, 12 edges or sides of equal length. Volume of a cube with side length 1 unit is 1 cubic unit. Such cube is called a "unit cube," and used as unit of volume.

Volume of a cube

A cube has all sides are of equal length,

So, Volume of a cube

= side x side x side = S x S X S cu units

Or

= length x length x length = l x l x l cu units

Or

= area x side cu units (And we know that area of square = side x side)

Volume of a cuboid

Similarly, a cuboid is a solid box whose every surface is a rectangle of same area or different areas. A cuboid has length, breadth and height.

So, Volume of a cuboid

= length x breadth x height = l x b x h cubic units

Or

= area of one surface x height cu units (as area =l x b)

Note: In a cuboid, when the length, breadth and height are of different units, convert them to a same unit and solve.

Volume by counting cubes

Step: Find out the number of unit cubes that form the solid or fill up the entire space occupied by the given solid such as cuboid and cube.

Example:

Take an empty cube shaped box of edge 4 cm with open top. Now fit cubes of edges 1 cm in it.

How many cubes of 1 cm can be fitted into cube box of edge 4 cm?

From the observation, it is clear that 64 such cubes will fit into it. So the volume of the box will be equal to the volume of 64 unit cubes together.

Therefore, the volume of the cube = 64 cu cm

Note that 64 = 4 × 4 × 4

Illustration 1: Find the volume of a cube of side 8 cm.

Solution: Volume of cube = l x l x l = 8 x 8 x 8 = 512 cu cm

Illustration 2: Find the volume of a cuboid of dimensions 16 cm x 10 m x 6 cm.

Solution: Volume of cuboid = l x b x h = 16 x 10 x 6 = 960 cu cm

Illustration 3: Find the volume of a cuboid of dimensions 18 cm x 30 mm x 15 cm.

Solution:

l = 18 cm,

b = 30 mm = 3 cm (10 mm =1 cm, 30/10 = 3 cm),

h = 15 cm

Volume of cuboid = l x b x h

= 18 x 3 x 15

= 810 cu cm

Illustration 4: Find the volume of a cuboid of dimensions 21 mm x 2 cm x 12 mm in cu. cm.

Solution:

10 mm = 1 cm

Therefore, 21 mm = 21/10 cm = 2.1 cm

And 12 mm = 12/10 = 1.2 cm

Length = 2.1 cm, breadth = 2 cm, height = 1.2 cm

Volume of cuboid = l x b x h

= 2.1 x 2 x 1.2

= 5.04 cu cm

Illustration 5: Find the number of cubical boxes of cubical side 5 cm, which can be accommodated in carton of dimensions 25 cm x 10 cm x 15 cm. Solution:

Volume of cubical box = side x side x side

= 5 x 5 x 5

= 125 cu cm

Volume of carton = 25 x 10 x 15

= 3750 cu cm

No. of boxes = Volume of carton/Volume of each box

= 3750/125

= 30