Curved surface area of cylinder is the measurement of outer area, where the extension of top and bottom portion wont be included.

If a rectangle revolves about one side and completes one full rotation, the solid thus formed is called a right circular cylinder. The above picture shows that how rectangle forms a right circular cylinder. In other words curved surface area is simply said as CSA

**CSA of cylinder = 2 π r h**

"r" and "h" stands for radius and height of cylinder.

A hollow cylinder is a three dimensional solid bounded by two parallel cylindrical surfaces and by two parallel circular bases cut out from two parallel planes by these two cylindrical surfaces.

**CSA of hollow cylinder = 2πh(R+r)**

R = external radius, r = internal radius and h = height

**Example 1 :**

A solid right circular cylinder has radius of 14 cm and height of 8 cm. Find its CSA.

**Solution :**

Radius of the cylinder (r) = 14 cm

Height of the cylinder (h) = 8 cm

Curved surface area of cylinder = 2Πrh

= 2⋅** **(22/7) ⋅ 14 ⋅ ** **8

= 704 sq.cm

Curved surface area of cylinder = 704 sq.cm

So, curved surface area of cylinder is 704 sq.cm

**Example 2 :**

Curved surface area and circumference at the base of a solid right circular cylinder are 4400 sq.cm and 110 cm respectively. Find its height and diameter.

**Solution :**

CSA of cylinder = 4400 sq.cm

Circumference of the base = 110 cm

2Πr = 110 ==> 2 ⋅ (22/7) ⋅ r = 110

r = 110 ⋅** **(1/2) ⋅ (7/22)

r = 17.5 cm

diameter = 2r = 2(17.5)

diameter = 35 cm

2 Π r h = 4400

110 ⋅ h = 4400

h = 4400/110

h = 40 cm

Height = 40 cm

Diameter of the cylinder = 35 cm

So, height and diameter of cylinder is 40 cm and 35 cm respectively.

**Example 3 :**

A mansion has 12 right cylindrical pillars each having radius 50 cm and height 3.5 m. Find the cost to paint the curved surface of pillars at $ 20 per square meter.

**Solution :**

The pillars of the mansion are in the shape of cylinder

Radius = 50 cm ==> 0.5 m

Height = 3.5 m

CSA of one pillar = 2 ⋅ (22/7) ⋅ 0.5 ⋅ 3.5

= 2 ⋅ 22 ⋅ 0.5 ⋅ 0.5 ==> 11 m^{2}

CSA of 12 pillars = 12 ⋅ 11

= 132 m^{2}

Cost to paint per m^{2} = $ 20

Total cost = 20 ⋅ 132

= $ 2640

Hence, total cost of painting 12 pillars is $ 2640

**Example 4 :**

The total surface area of a solid right circular cylinder are 231 cm². Its curved surface area is two thirds of the total surface area. Find the curved surface area if cylinder.

**Solution :**

Curved surface area = (2/3) ⋅ Total surface area

2 Π r h = (2/3) ⋅ 231

2 Π r h = 2 ⋅ 77

2 Π r h = 154

Hence, curved surface area of cylinder is 154 cm^{2}

**Example 5 :**

The total surface area of a solid right circular cylinder is 1540 cm². If the height is four times the radius of the base, then find the CSA of cylinder.

**Solution :**

Total surface area of cylinder = 1540 cm²

CSA of cylinder + top area + bottom area = 1540 cm²

2 Π r (h + r) = 1540

h = 4 ⋅ radius of the base

h = 4 r

2 Π r (4r+r) = 1540

2 Π r (5r) = 1540

2 ⋅ (22/7) ⋅ 5 r^{2} = 1540

= 1540 ⋅ (1/2) ⋅ (7/22) ⋅ (1/5)

= (1540 ⋅ 7)/(2 ⋅ 22 ⋅ 5)

= (1540 ⋅ 7 )/(2 ⋅ 22 ⋅ 5)

r^{2} = 49

r = √(7⋅7)

r = 7 cm

Curved surface area of cylinder :

= 1540 - 2 ⋅ (22/7) ⋅ 7 ⋅ 7

= 1540 - 308

= 1232 cm^{2}

Hence, CSA of cylinder = 1232 cm^{2}.

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