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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             (∵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.

⇒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 cuboid . We will also solve some examples using these formulae.

Did You Know?

The word ‘cuboid’ is made up of ‘cube’ and ‘-oid’ (which means ‘similar to’). So, a cuboid indicates something that is similar to a cube. A cuboid is also called a ‘rectangular prism’ or a ‘rectangular parallelepiped’.

Formulae for the Surface Area of a Cuboid

Consider a cuboid of length l, breadth b and height h.

The formulae for the surface area of this cuboid are given as follows:

Lateral surface area of the cuboid = 2h (l + b)

Total surface area of the cuboid = 2 (lb + bh + hl)

Here, lateral surface area refers to the area of the solid excluding the areas of its top and bottom surfaces, i.e., the areas of only its four standing faces are included. Total surface area refers to the sum of the areas of all the faces.

Did You Know?

Two mathematicians named Henri Lebesgue and Hermann Minkowski sought the definition of surface area at around the twentieth century.

Know Your Scientist

Henri Lebesgue (1875−1941) was a French mathematician who is famous for his theory of integration. His contribution is one of the major achievements of modern analysis which greatly expands the scope of Fourier analysis. He also made important contributions to topology, the potential theory, the Dirichlet problem, the calculus of variations, the set theory, the theory of surface area and the dimension theory.

 

 

 

 

Hermann Minkowski (1864−1909) was a Polish mathematician who developed the geometry of numbers and made important contributions to the number theory, mathematical physics and the theory of relativity. His idea of combining time with the three dimensions of space, laid the mathematical foundations for Albert Einstein’s theory of relativity.

 

 

 

Example Based on the Surface Area of a Cuboid

Did You Know?

The concept of surface area is widely used in chemical kinetics, regulation of digestion, regulation of body temperature, etc.

Formulae for the Surface Area of a Cube

Consider a cube with edge a.

The formulae for the surface area of this cube are given as follows:

Lateral surface area of the cube = 4a2

Total surface area of the cube = 6a2

Here, lateral surface area refers to the area of the solid excluding the areas of its top and bottom surfaces, i.e., the areas of only its four standing faces are included. Total surface area refers to the sum of the areas of all the faces.

Did You Know?

A cube can have 11 different nets. Cubes and cuboids are 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 al…

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