[Maths Class Notes] on Volume of a Cylinder Pdf for Exam

A cylinder is a closed solid having parallel circular bases attached by a curved surface. It is a three-dimensional shape with two circular ends called bases which are congruent and parallel to each other. The surface that connects the two circular ends is basically a rectangle.

 

Height is the perpendicular distance between two circular surface and radius of a cylinder is basically the radius of the circular base.

 

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A cylinder can be viewed as:

1. Elongated circle made by circles kept one above the other. Examples- Cylinder made out of bangles.

2. Combination of two circles and a quadrilateral (square, rectangle, etc) 

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A cylinder can be hollow or solid. A hollow cylinder is empty from inside whereas a solid cylinder is packed up with the same material. Layers of circular surfaces can be extracted from a solid cylinder if it is dissected cross-sectionally.

 

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The volume of a Cylinder

The volume of a cylinder is the space occupied by a cylinder. For a hollow cylinder, it is the space enclosed inside a cylinder and for solid cylinder, it is the amount of material the cylinder is composed of.

How to Calculate Volume of a Cylinder of Unknown Height and Radius

Take a cylinder of unknown radius, a container, a trough and a measuring cylinder.

Arrange the apparatus as shown below. Fill the container with water up to the brim and then carefully place the cylinder in the container. This will make some water flow out of the container which will get collected in the trough. Pour the water from trough to measuring cylinder to find out the amount of water displaced by the cylinder.

 

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The amount of displaced water is equal to the space occupied by the cylinder.

The Volume of Cylinder = Amount of Displaced Water

Cylindrical Volume Formula Of Known Height and Radius.

As discussed earlier, a cylinder can be seen as circles kept one above the other. If the breadth of the circle is of unit length then the height of the cylinder will be equal to the number of circles. The cylinder volume formula is shown below:

Volume = length x breadth x height

Volume = area of surface x height                       ———————–(i)

Where area of surface= length x breadth

 

Now for a cylinder, the area of the surface is equal to the area of the base (circle)

The volume of c formula = area of circular base x height

                                             = πr² × h

Therefore,

The Equation for Volume of a Cylinder = πr²h.

r= radius of the circular base

h= the perpendicular distance between two circular bases.

The Volume of a Hollow Cylinder of Known Height and Radius.

Let us suppose a hollow cylinder of outer radius R, inner radius r and height h. The hollow cylinder is actually the result of a smaller cylinder cut out from the bigger cylinder.

The formula for calculating the volume of a cylinder (hollow): 

= Volume of the bigger solid cylinder – Volume of the smaller solid cylinder.

= πR²h – πr²h

= πh (R² – r²)

Therefore, the cylinder formula = πh (R² – r²) cubic units.

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The Conversion of the Volume of a Cylinder 

Unit of the volume is cubic centimetre. Thousand cubic centimetres is equal to one litre. Thus, to convert a value from litre to cm³, 1000 is multiplied to the number. Similarly, to convert cm³ to litre, the number is divided by 1000. 

 

1 Litre = 1000 cubic cm or cm³

1 cm³ = 1/1000  litres

Solved Examples:

1) A cylinder has a base of radius 14 cm and the height of 10 cm. Find the volume.

Solution:

2) If the base of a cylinder is of area 30cm² and height is 6 cm then find the volume of the cylinder.

Solution:

Volume = Area of the base x height of a cylinder

               = 30 cm² x 6 cm

              = 180 cm³