Surface Area of a Cylinder: Understanding Formula, Curved and Total Area with Examples

The surface area of a cylinder is a fundamental concept in geometry that quantifies the total area occupied by all its external surfaces. The surface area of a cylinder refers to the total area covered by its outer surfaces, including the curved surface and the two circular bases. It is an important concept in geometry, widely used in solving real-world problems involving pipes, tanks, and cylindrical containers. A clear understanding of the formulas for total surface area (TSA) and curved surface area (CSA) of a cylinder enables accurate measurement. In this guide, you will learn about the formulas, methods, and step-by-step examples related to the surface area of a cylinder.

Table of Contents

• What is the Surface Area of Cylinder?
• Curved Surface Area (CSA) of a Cylinder
• Total Surface Area (TSA) of a Cylinder
• Derivation of the Surface Area Formula
• How to Calculate the Surface Area of Cylinder: Step by Step
• Solved Examples on Surface Area of Cylinder
• Differences Between CSA and TSA of Cylinder
• Practice Questions on Surface Area of Cylinder

What is the Surface Area of Cylinder?

A cylinder is a three-dimensional solid shape that has two flat circular ends (called bases) and one curved surface connecting them. Both circular bases are identical and parallel to each other.

The surface area of a cylinder is the total area of all the surfaces that make up the outside of the cylinder. There are exactly three surfaces on a cylinder:

• The top circular base

• The bottom circular base

• The curved (lateral) surface connecting them

Depending on whether you include both circles or just the curved part, you get two different types of surface area: total surface area (TSA) and curved surface area (CSA)

Curved Surface Area (CSA) of a Cylinder

The curved surface area (CSA) of a cylinder is the surface area covered by its curved surface only. The curved surface area is also called the lateral surface area. It is the area of just the tube part of the cylinder, with no tops or bottoms.

CSA of Cylinder Formula:

Let r be the radius of the circular base and h be the height of the cylinder. The value of π (pi) is taken as 22/7 or 3.14159…

Curved Surface Area of Cylinder, CSA = 2πrh sq. units

where r = radius and h = height of the cylinder

Total Surface Area (TSA) of a Cylinder

The total surface area of the cylinder (TSA of the cylinder) is obtained by adding the area of the two bases and the area of the curved surface. Since each circular base has an area of πr², and there are two of them:

Area of two bases = πr² + πr² = 2πr²

Adding the curved surface area: TSA = 2πrh + 2πr²

Total Surface Area of Cylinder Formula:

TSA = 2πrh + 2πr² = 2πr(r + h) sq. units

where r = radius and h = height of the cylinder

Derivation of the Surface Area Formula

Step 1: Unroll the cylinder.

Imagine you have a cylinder and you cut it along one vertical line and unroll it. The curved surface flattens into a rectangle.

Step 2: Calculate Each Part

The rectangle's length = circumference of the circle = 2πr

The rectangle's breadth = height of the cylinder = h

∴ Area of curved surface = 2πr × h = 2πrh

Area of each circular base = πr²

Area of both circular bases = 2πr²

Step 3: Add Everything Together

TSA = Curved Surface + Two Circular Bases

TSA = 2πrh + 2πr²

TSA = 2πr(r + h) 

How to Calculate the Surface Area of Cylinder: Step by Step

Follow the steps given below to find the surface area of a cylinder:

• Identify what you're asked to find. Is it the CSA (curved/lateral surface area) or the TSA (total surface area)?

• Note down the given values: radius (r) and height (h). If diameter is given, remember r = d/2.

• Substitute the values into the formula: CSA = 2πrh or TSA = 2πr(r + h).

• Use π = 22/7 or 3.14

• Calculate and write the unit: the answer must always be in square units (cm², m², etc.).

Solved Examples on Surface Area of Cylinder

Example 1: Find the curved surface area of a cylinder whose radius is 7 cm and height is 10 cm. (Take π = 22/7)

Solution: Given, r = 7 cm,  h = 10 cm

CSA = 2πrh

Substituting values in the formula

CSA = 2 × (22/7) × 7 × 10 = 2 × 22 × 10 = 440 cm²

Therefore, the curved surface area of cylinder = 440 cm²

Example 2: A cylindrical tank has a radius of 5 m and a height of 12 m. Find its total surface area. (π = 3.14)

Solution: Given, r = 5 m,  h = 12 m

TSA = 2πr(r + h)

r + h = 5 + 12 = 17 m

TSA = 2 × 3.14 × 5 × 17 = 6.28 × 85 = 533.8 m²

Therefore, the total surface area of cylinder = 533.8 m²

Example 3: The diameter of a cylindrical pillar is 28 cm, and its height is 4 m. Find the cost of painting its curved surface at ₹12 per m². (π = 22/7)

Solution: Given Diameter = 28 cm ⇒ r = 14 cm = 0.14 m;  h = 4 m

CSA = 2πrh

CSA = 2 × (22/7) × 0.14 × 4 = 2 × 22/7 × 0.56 = 2 × 1.76 = 3.52 m²

Cost of painting per m² = ₹12

⇒ Cost of painting 3.52 m²  = 3.52 × ₹12 = ₹42.24

Therefore, the cost of painting the curved surface of cylindrical pillar = ₹42.24

Example 4: The curved surface area of a cylinder is 264 cm² and its height is 14 cm. Find the radius. (π = 22/7)

Solution: Given, CSA = 264 cm²,  h = 14 cm

CSA = 2πrh

264 = 2 × (22/7) × r × 14 

⇒ 264 = 88r  

⇒  r = 264/88 = 3 cm

Radius of the cylinder = 3 cm

Example 5: A closed cylindrical water tank made of steel has a diameter of 1.4 m and a height of 2.1 m. Find the total surface area of the tank to determine how much steel sheet is needed. (π = 22/7)

Solution: Given: Diameter = 1.4 m ⇒ r = 0.7 m;  h = 2.1 m

TSA = 2πr(r + h)

r + h = 0.7 + 2.1 = 2.8 m

TSA = 2 × (22/7) × 0.7 × 2.8 = 2 × 22 × 0.1 × 2.8 = 2 × 6.16 = 12.32 m²

Steel required to make a closed cylindrical water tank of diameter 1.4m = 12.32 m²

Differences Between CSA and TSA of a Cylinder

Practice Questions on Surface Area of Cylinder

1. Find the CSA of a cylinder with radius = 3.5 cm and height = 10 cm.

2. The total surface area of a closed cylinder is 1628 cm², and its height is 14 cm. Find the radius.

3. A hollow cylindrical pipe is 21 dm long. Its outer diameter is 10 cm and inner diameter is 6 cm. Find the total surface area. 

4. How many square metres of canvas is required to make a cylindrical tent of radius 7 m and height 10 m? 

5. If the CSA of a cylinder is 1320 cm² and the radius is 10.5 cm, find the height of the cylinder.

6. A cylindrical container is open at the top. If its radius is 7 cm and height is 11 cm, find the area of the material used (excluding the top).

7. Find the height of a cylinder whose curved surface area is 154 cm² and radius is 3.5 cm.

8. A cylindrical water tank has a radius of 4 m and a height of 5 m. Find the cost of painting its curved surface at ₹20 per m².

9. A closed cylindrical drum has a radius of 35 cm and a height of 70 cm. Find the total surface area of the drum.

10. A company manufactures cylindrical cans of radius 7 cm and height 10 cm. Find the sheet metal required to make one can (closed).