TSA = 2πr(r + h). Worked example with r=3, h=10.

Surface Area of a Cylinder — Formula and Example

Question

A solid cylinder has radius r = 3 cm and height h = 10 cm. Find its Total Surface Area (TSA).

Solution — Step by Step

The TSA of a cylinder covers three faces: the curved surface plus two circular bases.

\text{TSA} = 2\pi r h + 2\pi r^2 = 2\pi r(r + h)

The factored form 2πr(r + h) is cleaner to use in calculations — only one multiplication by 2πr.

From the problem: r = 3 cm, h = 10 cm.

Substitute directly — don’t round π to 3.14 until the last step, or rounding errors compound.

\text{TSA} = 2\pi \times 3 \times (3 + 10)

= 2\pi \times 3 \times 13

= 78\pi \ \text{cm}^2

Using π ≈ 3.14:

\text{TSA} = 78 \times 3.14 = 245.04 \ \text{cm}^2

TSA = 245.04 cm²

Why This Works

A cylinder has exactly three surfaces. The two circular ends each have area πr², giving 2πr² together. The curved surface, if you “unroll” it, becomes a rectangle — its width is the circumference 2πr and its height is h, so the curved surface area is 2πrh.

Adding both parts: 2πr² + 2πrh = 2πr(r + h). The factored form isn’t just algebra — it tells you that both terms share the factor 2πr, which is the circumference of one circle.

This formula appears as a standard derivation question in CBSE Class 8 and Class 9. Understanding the “unrolling” argument guarantees you can re-derive it even if you forget the formula under exam pressure.

Alternative Method (Using Curved + Flat Separately)

Some students prefer computing the two parts separately before adding.

Curved Surface Area (CSA):

\text{CSA} = 2\pi r h = 2 \times 3.14 \times 3 \times 10 = 188.4 \ \text{cm}^2

Area of two circular bases:

2\pi r^2 = 2 \times 3.14 \times 3^2 = 2 \times 3.14 \times 9 = 56.52 \ \text{cm}^2

TSA = 188.4 + 56.52 = 244.92 cm²

The tiny difference (244.92 vs 245.04) is a rounding artifact from using π ≈ 3.14 at different stages. If the question asks for an answer in terms of π, always leave it as 78π cm² — that’s the exact answer.

When the answer choices in MCQs use π, keep your answer in the form kπ until the very end. Premature rounding is the #1 source of “almost right” answers.

Common Mistake

Confusing TSA with CSA. The Curved Surface Area is just 2πrh = 188.4 cm². Many students write this as the final answer and lose marks. TSA includes the two circular caps. Read the question: “total surface area” means all surfaces. If the cylinder is open at one end (like a pipe), you’d add only one πr² — the question will specify this.

\text{CSA} = 2\pi r h

\text{TSA} = 2\pi r(r + h)

\text{Volume} = \pi r^2 h

Units: if r and h are in cm, area is in cm², volume in cm³.

Question

Solution — Step by Step

Why This Works

Alternative Method (Using Curved + Flat Separately)

Common Mistake

Try These Next