Mensuration

Area of Trapezium

Consider a container, which is cylindrical in shape. Let us consider that 10 litres of milk can be stored in this container.

If the container is half-filled with milk, can we find the quantity of milk in the container?

Yes, we can find it. When the container is half-filled with milk, then the quantity of milk in the container is 5 litres.

Here, the quantity of milk in the container is the volume of the milk which is 5 litres.

And the container can store a maximum of 10 litres of milk, which is the capacity of the container.

If the container is completely filled with milk, then

Capacity of container = volume of milk = 10 litres

Thus we can say that,

“Volume is the amount of space occupied by an object, while capacity refers to the quantity that a container holds".

The units of volume of solid material are cm³, m³, dm³ etc and the unit of volume of liquid and capacity is litre.

Let us discuss some examples based on volume and capacity.

Example 1:

A cubical container has each side measuring 20 cm. The container is half-filled with water. Metal stones are dropped in the container till the water comes up to the brim. Each stone is of volume 10 cm³. Calculate the number of stones and the capacity of the container.

Solution:

We know that volume of cube = (side) ³

∴ Volume of cubical container = (20)³ cm³

                                                  = 8000 cm³

∴ Capacity of container = 8000 cm³

                                        = 8 litres             ($\because$1 litre = 1000 cm³)

The container is half-filled with water.

∴ Volume of water in the container = 4 litres

and,  volume of metal stones = 4 litres

                                               = 4 × 1000 cm³

                                               = 4000 cm³

Volume each metal stone = 10 cm³

∴ Number of stones =

                                  = 400 stones

Thus, the capacity of the container is 8 litres and the number of stones is 400.

Example 2:

An oil tank is in the form of a cuboid whose dimensions are 60 cm, 30 cm, and 30 cm respectively. Find the quantity of oil that can be stored in the tank.

Solution:

It is given that

Length (l) = 60 cm

Breadth (b) = 30 cm

Height (h) = 30 cm

∴ Quantity of oil = Capacity of tank

                            = l × b × h

                            = 60 × 30 × 30

                            = 54,000 cm³

We know that,

1 litre = 1000 cm³

∴ Quantity of oil that can be stored in the tank = 54 litres

Example 3:

Water is pouring in a cubical reservoir at a rate of 50 litres per minute. If the side of the reservoir is 1 metre, then how much time will it take to fill the reservoir?

Solution:

Side of reservoir = 1 m

∴ Capacity of reservoir = 1 m × 1 m × 1 m

                                       = 1 m³

We know that,

1 m³ = 1000 litres

∴ Capacity of reservoir = 1000 litres

50 litres of water is filled in 1 minute.

1 litre of water is filled in.

$\Rightarrow$1000 litres of water will be filled in =

                                                             = 20 min

∴ Thus, the reservoir is filled in 20 minutes.

Example 4:

Orange juice is available in two packs − a tin cylinder of radius 2.1 cm and height 10 cm and a tin can with rectangular base of length 4 cm, width 3 cm, and height 12 cm. Which of the two packs has a greater capacity?

Solution:

For tin cylinder,

Radius (r) = 2.1 cm

And, height (h) = 10 cm

Capacity of cylinder = πr²h

= 138.60 cm³

For tin can with rectangular base,

Length (l) = 4 cm

Width (b) = 3 cm

And height (h) = 12 cm

Capacity of tin can = l × b × h

= 144 cm³

Therefore, the tin can with a rectangular base has greater capacity than the tin cylinder.

Surface Areas of a Cube and a Cuboid

We give gifts to our friends and relatives at one time or another. We usually wrap our gifts in nice and colourful wrapping papers. Look, for example, at the nicely wrapped and tied gift shown below. 

Clearly, the gift is packed in box that is cubical or shaped like a cube . Suppose you have a gift packed in a similar box. How would you determine the amount of wrapping paper needed to wrap the gift? You could do so by making an estimate of the surface area of the box. In this case, the total area of all the faces of the box will tell us the area of the wrapping paper needed to cover the box. 

Knowledge of surface areas of the different solid figures proves useful in many real-life situations where we have to deal with them. In this lesson, we will learn the formulae for the surface areas of a cube and a cuboid . We will also solve some examples using these formulae.