Surface Area of Cylinders: Formula, CSA & TSA with Examples

A cylinder is the shape of a tin can, a water pipe or a battery: two circular ends joined by a curved surface that wraps around them. If you unroll that curved surface, it flattens into a rectangle whose sides are the circumference of the circle (2πr) and the height of the cylinder (h). This unrolling trick is the key to the cylinder's surface-area formulas.

Cylinders appear in packaging, plumbing, engines and storage tanks. Whether a question asks for the curved surface area (CSA) — the label around a can — or the total surface area (TSA) — the whole tin including its lid and base — the radius and height are the only two numbers you need. Once these two formulas are comfortable, questions about pipes, tanks and containers become routine.

Key Concepts

• Radius (r): the radius of the circular base.
• Height (h): the vertical distance between the two circular ends.
• Curved surface area (CSA): the area of the side that wraps around, not counting the two circular ends.
• Total surface area (TSA): the CSA plus the areas of the two circular ends.
• Areas are always measured in square units (cm², m²).

Formulas

• CSA: 2πrh — the unrolled rectangle of width 2πr and height h.
• TSA: 2πrh + 2πr² = 2πr(h + r) — curved area plus both circular ends.
• Hollow cylinder or pipe: outer CSA plus inner CSA = 2πRh + 2πrh, plus the two annular ring areas at the ends, π(R² − r²) each, to get the total.
• These formulas assume a right circular cylinder, whose axis is perpendicular to the base.

Worked Example

Find the CSA and TSA of a cylinder with radius 7 cm and height 20 cm. Use π = 22 ÷ 7.

1. CSA = 2πrh = 2 × (22 ÷ 7) × 7 × 20.
2. The 7s cancel: 2 × 22 × 20 = 880. The CSA is 880 cm².
3. Each circular end has area πr² = (22 ÷ 7) × 49 = 154 cm², so two ends give 308 cm².
4. TSA = 880 + 308 = 1188. The total surface area is 1188 cm².
5. Check with the combined formula: 2πr(h + r) = 2 × (22 ÷ 7) × 7 × 27 = 44 × 27 = 1188, which matches.

Common Misconceptions

• Confusing CSA with TSA: a label around a can is CSA (2πrh); the metal to make the whole can is TSA (2πr(h + r)). The wording of the question decides which one.
• Using the diameter as the radius: if a pipe's diameter is 14 cm, the radius is 7 cm — halve it before substituting.
• Forgetting one of the two circular ends: the TSA needs 2πr², not just πr². Dropping an end is a frequent slip.

Real-World Applications

• Printing labels for cans, jars and bottles (the CSA of the label).
• Painting or galvanising water pipes and storage tanks.
• Calculating sheet metal for cylindrical containers and dustbins.
• Estimating the insulation needed around hot-water pipes.

Unroll the curved surface into a rectangle and the formulas become obvious: 2πrh for the curve, plus two circles of πr² each, giving 2πr(h + r) in total. Halve any diameter first, count both ends, and always write the answer in square units.