The solid enclosed by this surface and by two planes perpendicular to the axis is called a cylinder. The cylinder has a round base and a perpendicular height. Surface area of a cylinder is the sum of surface area of each end of the cylinder and the surface area of the sides. The surface area of a cylinder depends on whether a cylinder is without a top or bottom, or can, which has both top and bottom.

The surface area of a cylinder whose circular base has radius 'r' and whose height is 'h' is SA = 2$\pi$r(r + h).

## Surface Area of a Cylinder Formula

The area of the surfaces of the cylinder is called surface area of the cylinder. The surface area of a cylinder is represented by square units like square centimeter, square millimeter and so on. Surface area, SA, of the side of the cylinder is the circumference of the circular base times the height and SA of the bases of the cylinder is the twice the area of the circle, with radius 'r'.

Formula:

The Surface Area of the Cylinder can be written as,

SA= 2$\pi r$(r + h)

where,
SA= Surface area of a cylinder
r = Radius of a cylinder
h = Height of a cylinder.

## Find the Surface Area of a Cylinder

To find the surface area of a cylinder, find the surface area of each end and each side of the cylinder.

Surface area of each end:
Since each end of a cylinder is a circle,

=> Surface area of each end = $\pi r^2$
where r is the radius of the circular end.

There are two circular ends so their combined surface area,

=> SA = $\pi r^2$ + $\pi r^2$ = 2$\pi r^2$. ........................(1)

And

Surface area of the side:
The length of the rectangle = Circumference of the base of a cylinder

= 2$\pi$r

=> Surface area of the side is the circumference times the height,

=> SA = 2 $\pi$ r h ...........................(2)
where r is the radius and h is the height of the side.

Combining (1) and (2)

The entire formula for the surface area of a cylinder = 2$\pi r^2$ + 2 $\pi$ r h

## Lateral Surface Area of a Cylinder

The lateral surface area of the cylinder is the area of the cylinder surface that does not include the circular ends. Lateral surface area of a cylinder is directly proportional to the radius and the height of the cylinder.

Formula:

Lateral surface area of a cylinder = 2 $\pi$ r h.

Where, r is radius and h is height of a cylinder.

Example:
Find the lateral surface area of right circular cylinder, if its base has a radius of 3 cm and its height is 12 cm.

Solution:
Step 1:

Radius of a cylinder = 3 cm
Height of a cylinder = 12 cm

Step 2:

Lateral surface area of a cylinder (SA) = 2 $\pi$ r h

=> SA = 2 * $\frac{22}{7}$ * 3 * 12

= 226.3

Hence the lateral surface area of a cylinder is 226.3.

## Total Surface Area of a Cylinder

Total surface area of a cylinder is the sum the surface area of each end and the surface area of the sides of the cylinder. The total surface area (TSA) includes the area of the circular top and base, as well as the curved surface area (CSA) of the cylinder.

TSA = Area of the lid + Area of the base + Curved surface area

= $\pi r^2$ + $\pi r^2$ + 2$\pi$ rh

= 2$\pi r^2$ + 2$\pi$ rh

= 2$\pi$r(r + h)

Formula:

Total Surface Area of a Cylinder = 2 $\pi$ r(r + h).

Where, r is radius and h is height of a cylinder.

Example:
Find the total surface area of a cylindrical tin of radius 6 cm and height 9 cm.

Solution:
Step 1:
Radius of a cylinder = 6 cm
Height of a cylinder = 9 cm

Step 2:

Total surface area of a cylinder (TSA) = 2 $\pi$ r (r + h)

=> TSA = 2 * $\frac{22}{7}$ * 6(6 + 9)

= 12 * $\frac{22}{7}$ (15)

= 565.71

Hence the total surface area of a cylinder is 565.71 cm2 .