# Surface Area of a Cylinder

The surface area of a cylinder is the total space occupied by the flat circular bases of the cylinder and the curved surface. Simply, it is the sum of an area of a circle, since the base of the cylinder is a circle and the area of the curved surface which is a rectangle.

It is expressed in square units such as m2, cm2, mm2, and in2

A cylinder has 2 types of surface areas:

• Curved (Lateral) Surface Area
• Total Surface Area

## Formulas

### Lateral (Curved) Surface Area

The lateral or the curved surface area (LSA) of a cylinder is the area formed by the curved surface of the cylinder. It is thus the space occupied between the two parallel circular bases. The formula to calculate LSA is given below:

Let us solve some examples to understand the concept better.

Find the curved surface area of a cylinder whose diameter is 44 cm and height is 16 cm.

Solution:

As we know,
Radius (r) = d/2, here d = diameter
= 44/2 cm
= 22 cm
Now, as we know
Curved Surface Area (CSA) = 2πrh, here π = 3.141, r = 22 cm, h = 16
= 2 × 3.141 × 22 × 16
= 2211.26 cm2

The lateral surface area of a cylinder is 122 ft2. If the radius of the cylinder is 5 ft, calculate the height of the cylinder.

Solution:

As we know the curved surface area is the lateral surface area of the cylinder,
Curved Surface Area (CSA) = 2πrh, here CSA =122 ft2, π = 3.141, r = 5 cm
=> 122 = 2 × 3.141 × 5 × h
=> h = 122/2 × 3.141 × 5
=> h = 122/31.41
=> h = 3.8 ft

### Total Surface Area

The total surface area (TSA) of a cylinder is the sum of the curved surface area and the area of two circular bases. The formula to calculate TSA is given below:

Derivation

As we know, a cylinder has 2 flat surfaces which are usually circles, and a curved surface which is in the form of a rectangle.

Let the height of the cylinder be ‘h’, and the circular base has a radius ‘r’.  See the diagram below.

Now,

Total Surface Area (TSA) = Curved Surface Area (CSA) + (2 × Area of 1 circular base)

As we know,

Curved Surface Area (CSA) = 2πrh, and

Area of 1 circular base = πr2

∴ The area of 2 circular bases = 2πr2

Thus,

TSA = 2πrh + 2πr2

TSA = 2πr(r + h)

Thus, the equation to calculate TSA of a cylinder is TSA = 2πr(r + h)

Let us solve some examples to understand the concept better.

Find the total surface area of a cylinder whose radius is 8 cm and height is 12 cm.

Solution:

As we know,
Total Surface Area (TSA) = 2πr(r + h), here π = 3.141, r = 8 cm, h = 12 cm
= 2× 3.141 × 8(8 + 12)
= 2× 3.141 × 8 × 20
= 1005.12 cm2

Find the height of a cylinder if its total surface area is 2442 inand the radius is 12 in.

Solution:

As we know,
Total Surface Area (TSA) = 2πr(r + h), here TSA = 1224 in2, π = 3.141, r = 12 in
=> 2442 = 2 × 3.141 × 12(12 + h)
=> 2442 = 75.384(12 + h)
Divide both sides by 75.384, we get
=> 32.39 = 12 + h
Subtracting 12 from both sides, we get
h = 32.39 in

Calculate the radius of a cylinder whose total surface area is 222.46 square feet, and the height is 4 feet.

Solution:

As we know,
Total Surface Area (TSA) = 2πr(r + h), here TSA = 222.46ft2, π = 3.141, h = 4 ft
=> 222.46 = 2 × 3.141 × r(r + 4)
=> 222.46 = 6.282r(r + 4)
By distributive property of multiplication on the R.H.S, we get
=> 222.46 = 6.282r2 + 25.128
Dividing each term by 6.282, we get
=> 35.412 = r2 + 4
=> r2 = 31.412
=> r = 5.6 ft

## Surface Area of an Open Cylinder

An open cylinder is a cylinder without a top. Thus the surface area of an open cylinder can be calculated by finding the area of one base and the curved surface.

The formula to calculate the surface area of an open-top cylinder is given below:

Surface Area (A) = πr(rh + r), here π = 22/7 = 3.141, r = radius, h = height

Let us solve an example to understand the concept better.

Find the surface area of an open cylinder with a height of 10 m and a radius of 18 m.

Solution:

As we know,
Surface Area (A) = πr(rh + r), here π = 3.141, r = 18 m, h = 10 m
= 3.141 × 18(18 × 10 + 18)
= 3.141 × 18 × 198
= 11194.52 m2