In this article provided formulas of Surface Area and Volume of a Right circular cylinder , Oblique Cylinder & Hollow Cylinder with examples

Contents

## Formulas for **Right circular cylinder**, **Oblique Cylinder & Hollow Cylinder**

**Cylinder Definition**

A cylinder is a solid which has both its ends in the form of a circle. Its dimensions are defined in the form of the radius of the base (r) and height ( h).

i.e The lateral surfaces are curved and ends are congruent circles

**Right circular cylinder**

A The line joining the centers of the circular bases is perpendicular to base, solid figure is called right circular cylinder.

**Surface Area & Volume of a ****Right circular cylinder**

Curved surface area of the right circular cylinder = *Perimeter of the base of the cylinder X height*

Perimeter of the base of the cylinder = Perimeter of circle with same radius = 2 π r

**Curved Surface Area of Cylinder = 2 π r h**

Total surface area of the right circular cylinder = Curved Surface Area of Cylinder + Base area + Top area

Base area = Top area = Area of the circle with same radius = π r^{2 }

= 2 π r h + π r^{2 }+ π r^{2 }

= 2 π r ( r + h)

**Total Surface Area of Cylinder = 2 π r ( r + h)**

Volume of a cylinder = *Area of the circular base X height* =* π r ^{2 }*X

*h*

**Volume of a Cylinder = π r ^{2 }h**

**Oblique Cylinder**

A The line joining the centers of the circular bases. The cylinder is sideways and the axis is not a right angle to the base, then it is called an oblique cylinder.

Its dimensions are defined in the form of the radius of the base (**r**) and lateral height or perpendicular height or altitude (** h**), Slant height (** a**), angle ( **x**)

**Surface Area & Volume of an **oblique cylinder

Perpendicular height of a Oblique Cylinder

**h = a sin x**

Lateral or Curved surface area of a Oblique Cylinder ( A_{a} )

**A _{a} = 2 π r a**

Total Surface Area of Oblique cylinder ( A_{s} )

*Total Surface Area of Cylinder = 2 π r ( r + a)*

Volume of Oblique Cylinder is the same as for the Right circular cylinder

*Volume of a Oblique Cylinder = π r ^{2 }h = π r^{2 }a sin x*

### Hollow Cylinder

A hollow cylinder is the figure in shape formed by just the lateral surface of the cylinder.

i.e A Hollow cylinder can be defined as, It is a solid bounded by two co-axial cylinders of the same height

Its dimensions are defined in the form of the radius of the base outer (**R**) , Inner (** r**) & height (** h**)

**Surface Area & Volume of a hollow cylinder**

Curved Surface of hollow cylinder = Outer surface area of the cylinder + Inner surface area of cylinder

= 2 π R h + 2 π r h

*Curved Surface of hollow cylinder = 2 π h ( R + r)*

Total surface area hollow cylinder = Outer surface area of the cylinder + Inner surface area of cylinder + Area of a base circular ring + Area of a top circular ring

= 2 π R h + 2 π r h + (π R^{2 }– π r^{2 } ) + (π R^{2 }– π r^{2 })

= 2 π h ( R + r) + 2 π ( R^{2 }– r^{2 })

*Total surface area hollow cylinder = 2 π ( R + r) (h + R – r)*

Volume of the material used for hollow cylinder = Volume of the cylinder with outer radius – Volume of the cylinder with inner radius

*Volume of the material for hollow cylinder = π R ^{2 }h – π r^{2 }h = π h ( R^{2} – r^{2 })*

### Examples on surface area and volume calculation of cylinder

**Example-1 :** In a cylinder volume (v) = 176 cm^{3} , h = 14 cm, then find the radius of cylinder

Solution: Let the radius of cylinder = r , Then

Volume of cylinder = 176 = π r^{2 }h

⇒ π r^{2 }(14) = 176

⇒ r^{2 }= 4

⇒ r^{ }= 2

**Example-2 **: 15 number of identical spheres are melted and converted into cylinder shape of 10 cm radius and 5.4 cm height is made. Then find the radius of sphere.

Solution: Let the radius of spheres = r

Total volume of spheres = Volume of resultant cylinder

⇒ 15 x (4/3) x π x r^{3 }= π x (10)^{2 }x 5.4

⇒ r^{3 }= 27

⇒ r^{ }= 3 cm

**Example-3: **The diameter of a garden roller is 0.7 m and it is 2 m long. Then find area covered by it in 10 revolution.

Solution: Garden roller covered in one revolution = Curved Surface Area of roller

Here dia of roller (D)= 0.7 m and long (h) = 2 m

i.e Lateral surface area of the garden roller = π D h = (22/7) x 0.7 x 2 = 4.4 m^{2}

For 10 revolutions = 10 x 4.4 = 44 m^{2}

**Example-4:** A hollow pipe long 20 cm, outer dia is 25 cm and thickness of the metal 1 cm. Find the total surface area of the pipe

Solution: Here radius of outer shell = R = 25/2 = 12.5 cm

Length = h = 14 cm

Thickness of pipe = R -r = 1 cm

Inner shell radius = r = 12.5 – 1 = 11.5 cm

Total surface area of the pipe = 2 π ( R + r) (h + R – r)

= 2 x ( 22/7) x ( 12.5 + 11.5 ) x ( 20 + 1) = 3168 cm^{2}

**Example-5:** The height of a metallic hallow cylinder is 14 cm and difference between its inner curved surface area and outer curved surface area is 44 cm^{2}. If the cylinder is made up of volume 99 cm^{3} metal. Find its inner & outer radius.

Solution: Let radius of outer shell = R & inner shell radius = r

Length = h = 14 cm

Here difference between its inner curved surface area and outer curved surface area is 44 cm^{2}

i.e ⇒ 2 π R h – 2 π r h = 44

⇒2 x (22/7) x 14 (R – r) = 44

⇒ R – r = 1/2 ————————- (Eq- 1)

Volume of cylinder = 99 cm^{3}

i.e

⇒ π h ( R^{2} – r^{2 }) = 99

⇒ π x 14 x ( R + r ) (R -r) = 99

From the equation – 1

⇒ R + r = 9/2 ——————– (Eq- 2)

Simplifying the eq-1 & eq-2

R = 5/2 cm & r = 2 cm

**Example-6: **A well of 14 m base radius is digged soil upto a depth of 10 m and from this soil material was made 7 m wide platform around the well. Find the height of platform

Solution: Here Well radius r*_{1}* = 10 m , Depth of well h

*= 10 m platform radius inner r = 14 m , outer radius = 21 m, height of the plat form = h*

_{1}⇒ Volume of well = Volume of plat form

*⇒ π r_{1}^{2 }h_{1} = π h ( R^{2} – r^{2 })*

*⇒* * π* x 14 x 14 x 10 =

*x h x (21*

**π***– 14*

^{2}*)*

^{2}*⇒* 14 x 14 x 10 = h x (21+14) x (21-14)

*⇒* h = 8

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