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" and "r" stands for external and internal radii of hollow cylinder and "h" stands for height.**

**Question 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 = 2 Π r h

= 2 **x** (22/7) **x** 14 **x** 8

= 704 sq.cm

Curved surface area of cylinder = 704 sq.cm

**Hence, Curved surface area of cylinder is 704 sq.cm**

**Question 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

2 Π r = 110 ==> 2 **x** (22/7) **x** r = 110

r = 110** x **(1/2)** x **(7/22) ==> r = 17.5 cm

diameter = 2 r

= 2 (17.5) ==> 35 cm

2 Π r h = 4400 ==> 110 **x**** ** h = 4400 ==> h = 4400/110

h = 40 cm

Height = 40 cm

Diameter of the cylinder = 35 cm

**Hence, height and diameter of cylinder are 40 cm and 35 cm respectively.**

**Question 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 **x** (22/7) **x** 0.5 **x** 3.5

= 2 **x** 22 **x** 0.5 **x** 0.5 ==> 11 m²

CSA of 12 pillars = 12 **x** 11 ==> 132 m²

Cost to paint per m² = $ 20

Total cost = 20 **x** 132

= $ 2640

**Hence, total cost of painting 12 pillars is $ 2640**

**Question 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)** x** Total surface area

2 Π r h = (2/3)** x** 231

2 Π r h = 2** x** 77

2 Π r h = 154

**Hence, curved surface area of cylinder is 154 cm²**

**Question 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 **x **radius of the base ==> h = 4 r

2 Π r (4 r + r) = 1540 ==> 2 Π r (5 r) = 1540

2 **x** (22/7) **x** 5 r² = 1540

= 1540** x** (1/2) **x** (7/22) **x** (1/5)

= (1540 **x **7)/(2** x** 22 **x** 5)

= (1540 **x **7)/(2** x** 22 **x** 5)

r² = 49 ==> r = √7**x **7 ==> r = 7 cm

Curved surface area of cylinder = 1540 - 2 **x** (22/7) **x** 7 **x** 7

= 1540 - 308 ==> 1232 cm²

**Hence, CSA of cylinder = 1232 cm²**

**Question 6 :**

The external surface area of a hollow cylinder is 540 Π cm².Its internal diameter is 16 cm and height is 15 cm. Find the curved surface area.

**Solution :**

External surface area of cylinder = 540 Π cm²

2 Π R h = 540 Π cm²

Internal radius (r) = 16/2 = 8 cm

height (h) = 15 cm

2 **x** Π **x** R **x** h = 540 Π ==> 2 **x** Π **x** R **x** 15 = 540 Π

= 540 Π** x** (1/2) **x** (1/Π) **x** (1/15) ==>270/(15) ==> R = 18

Curved surface area = 2 Π h (R + r) ==>2Π(15)(18+8) ==> 390Π cm²

**Curved surface area of the cylinder = 390 Π cm²**

**Question 7 :**

The external diameter of the cylindrical shaped iron pipe is 25 cm and its length is 20 cm. If the thickness of the pipe is 1 cm, find the curved surface area of the pipe.

**Solution :**

External radius of the pipe (R) = 12.5 cm

height of the pipe (h) = 20 cm

thickness of the pipe (w) = 1 cm

To find curved surface area of the cylinder we have to find the internal radius (r)

W = R - r ==> 1 = 12.5 - r ==> r = 12.5 - 1 ==> r = 11 .5 cm

Curved surface area = 2 Π h (R + r)

= 2 Π (20) (12.5 + 11.5) ==> 960 Πcm²

**Hence curved surface area of cylinder is 960 Πcm².**

**Question 8 :**

The radii of two right circular cylinders are in the ratio 3:2 and their heights are in the ratio 5:3. Find the ratio of their curved surface areas.

**Solution :**

Let r₁ , r₂ and h₁ , h₂ are radii and heights of first and second cylinders respectively.

Now we have to find the ratio of their curved surface areas

Curved surface area of cylinder = 2 Π r h

Curved surface area of first cylinder = 2 Π r₁ h₁

Curved surface area of second cylinder = 2 Π r₂ h₂

r₁ : r₂ = 3 : 2 ==> r₁/r₂ = 3/2 ==> r₁= 3r₂/2

h₁ : h₂ = 5 : 3 ==> h₁/h₂ = 5/3 ==> h₁= 5h₂/3

Ratios of curved surface area of two cylinders

2 Π r₁ h₁ : 2 Π r₂ h₂ ==> (3r₂/2) (5h₂/3) : r₂ h₂ ==> 5/2 : 1 ==> 5 : 2

**Hence, ratio of their curved surface areas is 5 : 2.**

**Question 9 :**

The diameter of the road roller of length 120 cm is 84 cm. If it takes 500 complete revolutions to level play ground, then find the cost of leveling it at the cost of $ 1.5 per square meter.

**Solution :**

**Radius of cylinder (r) = 84/2 => 42 cm**

**height of cylinder (h) = 120 cm**

**Area covered by road roller in one revolution = CSA of the road roller **

**Curved surface area of cylinder = 2 Π r h**

**= 2 x (22/7) ****x **** 42 ****x **** 120 ==> 31680 cm**²

Area covered by 500 revolutions = 500 x 31680 ==> 15840000 **cm**²

= 1584 **m**²

Cost of leveling per square meter = $ 1.5

Required cost = 1584 x 1.5 ==> $ 2376

**Hence, required cost is $ 2376.**

**Question 10 :**

The internal and external radii of of a hollow cylinder are 12 cm and 18 cm respectively. If its height is 14 cm, then find its curved surface area.

**Solution :**

External radius (R) = 18 cm

Internal radius (r) = 12 cm

height (h) = 14 cm

Curved surface area = 2 Π h (R + r)

= 2 x (22/7) (14) (18 + 12) ==> 2640 cm²

**Hence curved surface area of cylinder is 2640 cm².**

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