# Curved Surface Area Cylinder Formula

## Formula of Curved Surface Area Cylinder

A cylinder is a solid, three-dimensional shape with two parallel, circular bases and a Curved Surface Area Cylinder Formula at a fixed distance. One can also think of it as a stack of circular discs. You can also think of it as a stack of circular discs. Radius (r) is the distance between the axis and the outer Curved Surface Area Cylinder Formula, and height (h) is the distance between the two parallel circular bases. Cylinders include candles, water tanks, pipes, wells, etc.

### Surface Area of Cylinder – Introduction

Area is defined in geometry as the region covered by any shape in a plane.There are two types of surfaces in a cylinder: Curved Surface Area Cylinder Formula and the circular surface area formula. The area of both circular bases is the same.

It is possible to classify the surface area of the cylinder into two types:

1. CSA (curved surface area)
2. The total surface area (TSA)

### Surface Area of Cylinder

A Cylindrical surface consists of all points on all lines that are parallel to a given line and that pass through a fixed plane curve in a plane not parallel to the given line. In kinematics, a cylindrical surface is that which is traced out by a line, called the generatrix, that is not in the plane of the directrix, moving parallel to itself, and always passing through the directrix. An element of the cylindrical surface is any position of the generatrix.

A (solid) cylinder is a solid that is defined by two parallel planes and a cylindrical surface. A cylinder’s base is the region bounded by its cylindrical surface in either of its parallel planes. Cylinders have congruent bases. A right cylinder is one whose elements are perpendicular to the planes containing its bases, while an oblique cylinder is one whose elements are perpendicular to the planes containing its bases. Circular cylinders are those whose bases are discs (regions whose boundaries are circles). In some elementary treatments, cylinders are always circular in some elementary treatments.

Height (or altitude) is the distance between the bases of a cylinder.

By rotating a line segment about a parallel line, a cylinder of revolution is obtained. The cylinder of revolution is a right circular cylinder. The length of the generating line segment determines the height of a cylinder of revolution. It passes through the centres of both bases and is called the axis of the cylinder.

### Curved Surface Area (CSA) of Cylinder

Curved Surface Area Cylinder Formula is defined as the area of the curved surfaces after subtracting the areas of their circular bases. Also known as lateral surface area (LSA). In the case of a cylinder with a radius “r” and height “h,” the CSA is given by:

The curved surface area of the cylinder (Curved Surface Area Cylinder Formula) = 2rh sq. units.

### Total Surface Area (TSA) of Cylinder

Cylinders have two circular bases and the Curved Surface Area Cylinder Formula , which add up to their total surface area. The total surface area (TSA) of a cylinder is equal to the area of its two circular bases plus the area of its curved surface.

or,

TSA of cylinder = Curved Surface Area Cylinder Formula + Area of its two circular bases

= 2πrh + 2πr2 sq. units

So, TSA of cylinder = 2πr (h + r) sq. units

### Derivation of the Formula of Surface Area of Cylinder

Imagine covering a cylinder with coloured papers. Covering must be done with a minimum amount of paper. To cover the cylinder, take a rectangular sheet of paper whose length (l) is just enough to go around the cylinder and whose width (b) is equal to the height (h).

It should be noted that the length (l) of the rectangular sheet is equal to the circumference (r) of the circular base.

In the cylinder, the Curved Surface Area Cylinder Formula is determined by the area of the rectangular sheet.

Therefore, the area of the rectangular sheet of paper is equal to the Curved Surface Area Cylinder Formula

= 2πr × h

This means that the Curved Surface Area Cylinder Formula is equal to 2πrh sq. units.

r is the radius of the base, and h is its height.

Likewise, if the bottom and the top of the cylinder are to be covered with coloured papers, we need two circular regions, with radius r and area r2.

So, the total surface area of the cylinder equals the Curved Surface Area Cylinder Formula plus the area of the two circular regions.

= Curved Surface Area Cylinder Formula + πr2 + πr2

= 2πrh + 2πr2

Therefore, Total surface area of cylinder = 2πr (h + r) sq. units

Radius r and height h of the cylinder are its parameters.

Examples of Curved Surface Area Cylinder

Q.1. A cylindrical pillar has a diameter of 50 cm and a height of 7 m. Find the cost of painting the Curved Surface Area Cylinder Formula of the pillar at ₹12 per sq. mts.

Given, the height (h) of the cylindrical pillar is 7 meters

A circular base has a radius (r) equal to 50/2 cm = 25 cm = 0.25 m

The Curved Surface Area Cylinder Formula pillar is equal to 2πrh sq. units

= 2 ×(22/7)× 0.25 × 7

= 2 ×(22/7)× (25/100) × 7

= 11 sq. mts

Given, the cost of painting 1 sq. mts of area is ₹12.

∴ the cost of painting 11 sq. mts = 11 × 12 = ₹132

Therefore, the cost of painting the curved surface is ₹132.