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Right Circular Cylinder - Surface Areas and Volumes, CBSE, Class 10, Mathematics Notes | Study Additional Documents & Tests for Class 10 - Class 10

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RIGHT CIRCULAR CYLINDER
Solids like circular pillars, circular pipes, circular pencils, measuring jars, road rollers and gas cylinders, etc., are said to be in cylindrical shape.

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation
In mathematical terms, a right circular cylinder is a solid generated by the revolution of a rectangle about its sides.

Let the rectangle ABCD revolve about its side AB, so as to describe a right circular cylinder as shown in the figure. You must have observed that the cross-sections of a right circular cylinder are circles congruent and parallel to each other.

Cylinders Not Right Circular
There are two cases when the cylinder is not a right circular cylinder.

Case-I : In the following figure, we see a cylinder, which is certainly circular, but is not at right angles to the base. So we cannot say it is a right circular cylinder.

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

Case-II : In the following figure, we see a cylinder with a non-circular base as the base is not circular. So we cannot call it a right circular cylinder.

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

 

REMARK : Unless stated otherwise, here in this chapter the word cylinder would mean a right circular cylinder.
The following are definitions of some terms related to a right circular cylinder :

(i) The radius of any circular end is called the radius of the right circular cylinder.
Thus, in the above figure, AD as well as BC is a radius of the cylinder.

(ii) The line joining the centres of circular ends of the cylinder, is called the axis of the right circular cylinder.
In the above figure, the line AB is the axis of the cylinder. Clearly, the axis is perpendicular to the circular ends.

REMARK : If the line joining the centres of circular ends of a cylinder is not perpendicular to the circular ends, then the cylinder is not a right circular cylinder.

(iii) The length of the axis of the cylinder is called the height or length of the cylinder.

(iv) The curved surface joining the two bases of a right circular cylinder is called its lateral surface.

Formulae
For a right circular cylinder of radius = r units & height = h units, we have :

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

The above formulae are applicable to solid cylinders only.

Hollow Right Circular Cylinders

Solids like iron pipes, rubber tubes, etc., are in the shape of hollow cylinders.

A solid bounded by two coaxial cylinders of the same height and different radii is called a hollow cylinder.

  Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

Formulae
For a hollow cylinder of height h and with external and internal radii R and r respectively, we have :

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

 Ex.6 2.2 cu dm of brass is to be drawn into a cylindrical wire of diameter 0.50 cm. Find the length of the wire.
 Sol. 
Volume of brass = 2.2 cu dm = (2.2 × 10 × 10 × 10) cm3 = 2200 cm3. Let the required length of wire be x cm.
Then, its volume  Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation = 11200 cm = 112 m.

Hence, the length of wire is 112 m.

 Ex.7 A well with 14 m diameter is dug 8 m deep. The earth taken out of it has been evenly spread all around it to a width of 21 m to form an embankment. Find the height of the embankment.

Sol. Volume of earth dug out from the well Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation
Area of the embankmentMathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, PreparationMathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation
Height of the embankment = Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation = 53.3 cm.

Ex.8 The difference between the outside and inside surface of a cylindrical metallic pipe 14 cm long is 44 cm2. If the pipe is made of 99 cu cm of metal, find the outer and inner radii of the pipe.

Sol. Let, external radius = R cm and internal radius = r cm.

Then, outside surface Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation
Inside surface Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation
Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, PreparationMathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation
External volume Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation
Internal volume Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation
Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation
On dividing (ii) by (i), we get:

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation
Solving (i) and (iii), we get, R = 2.5 and r = 2.
Hence, outer radius = 2.5 cm and inner radius = 2 cm.

Ex.9 A solid iron rectangular block of dimensions 4.4 m, 2.6 m and 1 m is cast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.

Sol. Volume of iron = (440 x 260 x 100) cm3.

Internal radius of the pipe = 30 cm.

External radius of the pipe = (30 + 5) cm = 35 cm.

Let the length of the pipe be h cm.

Volume of iron in the pipe = (External volume) - (Internal volume)
Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, PreparationMathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation
Hence, the length of the pipe is 112 m.

Ex.10 A cylindrical pipe has inner diameter of 7 cm and water flows through it at 192.5 litres per minute. Find the rate of flow in kilometres per hour.

Sol. Volume of water that flows per hour = (192.50 × 60) litres = (192.5 × 60 × 1000) cm3.

Inner radius of the pipe = 3.5 cm.

Let the length of column of water that flows in 1 hour be h cm.
Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

Mathematics, Class X, Questions and Answers, Q and A, Important, CBSE, NCERT, Board Exam, Preparation

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