A cylinder is a three-dimensional figure with two round flat bases and one curved side. In other words, a closed solid has two parallel circular bases connected by a curved surface. It has no vertices. Let us now discuss the surface area of cylinder formula.

## Lateral or Curved Surface Area of Cylinder Formula

A cylinder is a rectangle with two circular bases. The faces of the cylinder are parallel and congruent circles. The cylinder has one curved surface. The height of the cylinder is the perpendicular distance between the base.

The lateral surface of an object is defined as the area of a curved surface, excluding the area of its base and top. In the lateral surface area of a cylinder, the radius is directly proportional to the height of the Cylinder.

**The Curved Surface Area of the Cylinder = \(2 \pi r h\)**

Where,

\(\pi\) | Pi, approximately 3.142 |

r | the radius of the cylinder |

h | height of the cylinder |

## Total Surface Area of Cylinder Formula

The surface area of the cylinder is the area of a curved surface of a cylinder including circular top and base.

**Area =\(2 \pi r (r + h)\)**

Where,

\(\pi\) | Pi, approximately 3.142 |

r | the radius of the cylinder |

h | height of the cylinder |

## Derivation of Surface Area of Cylinder Formula

### Curved Surface Area

Take a rectangle and roll it up, you end up with a cylinder. Therefore, if you want to compute the curved surface area of a cylinder, imagine unrolling it. The area of the rectangle is the width multiplied by the height. Here, the width of the rectangle is the height h of the cylinder, and the length of the rectangle is the circumference of the circle is 2 \pi r.

\(Area of rectangle = l × b = 2 \pi r × h\)

\(The Curved Surface Area of the Cylinder = 2 \pi r h\)

### Total Surface Area of a Cylinder

- Area of each circle can be found from the radius r of the circle. The area of a circle is \pi r
^{2}, so the combined area of the two circles is 2 \pi r^{2}. - The area of the rectangle is the breadth times height. The breadth of the rectangle is the height h of the cylinder, and the length of the rectangle is the distance around the end circles. This is the circumference of the circle is 2 \pi r. Thus, the rectangle area is 2 \pi r × h.

Combining these parts, we get the final formula:

Area = 2 π r^{2 }+ 2 π r h

By factoring 2 π r from each term, we can simplify this to

Area = 2 π r (r + h)

## Solved Examples

Q.1. Find the curved surface area of a cylinder of height 6 cm and the radius is 2 cm.

Solution:** **h = 6 cm; r= 2 cm

Area = 2 π r h

= 2 × 3.14 × 2 × 6

= 75.36 cm^{2}

Q.2. Find the total surface area of a cylindrical tin of radius 17 cm and height 3 cm.

Solution: Radius *r = *3 cm, Height *h* = 17 cm

Total surface area of tin = 2 π r (r + h)

= 2 × 3.14 × 3 (3 + 17)

= 18.84 × 20

= 376.2 cm^{2}

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