Surface area

Wrap a shoebox in gift paper and you instinctively know the paper has to cover every face — top, bottom, front, back, and both sides — with no gaps and no overlap wasted. That total covering is exactly what mathematicians call surface area: the sum of the areas of all the outer surfaces of a solid, measured in square units like $\text{cm}^2$ or $\text{m}^2$ . In Class 8, this idea is made precise for three solids only — the cube, the cuboid, and the right circular cylinder. Each has its own formula, built by adding up the areas of its flat or curved faces one by one. The cylinder brings in a second idea worth separating clearly: its curved surface area (CSA), the area of just the rolled-up side, versus its total surface area (TSA), which adds the two circular ends. Painters, tin-makers, and packagers all use these formulas daily — to quote a job, they first need to know exactly how much surface they are covering.

The core insight

Surface area adds up every outer face of a solid — for the cube and cuboid all faces are flat rectangles or squares, while for a cylinder the curved side (CSA) and the two circular ends are counted separately before being combined into TSA.

The core idea

A solid's surface area is built face by face, so the fastest way to find a formula is to count the distinct faces and add their areas.

Cuboid. A cuboid (length $l$ , breadth $b$ , height $h$ ) has 6 rectangular faces arranged in 3 identical pairs: the top and bottom ( $l \times b$ each), the front and back ( $l \times h$ each), and the two sides ( $b \times h$ each).

TSA_{cuboid} = 2(lb + bh + hl)

where: $l$ , $b$ , $h$ are the length, breadth, and height of the cuboid, all measured in the same unit.

Cube. A cube of edge $a$ is simply a cuboid where $l = b = h = a$ , so all 6 faces are identical squares of area $a^2$ .

TSA_{cube} = 6a^2

where: $a$ is the edge length of the cube.

Right circular cylinder. Peel the curved side off a cylinder (radius $r$ , height $h$ ) and flatten it out — it becomes a rectangle of width $h$ and length equal to the circle's circumference, $2\pi r$ . That rectangle is the curved surface.

CSA_{cylinder} = 2\pi rh

Add the two flat circular ends, each of area $\pi r^2$ , to get the total surface area of a fully closed cylinder.

TSA_{cylinder} = 2\pi rh + 2\pi r^2 = 2\pi r(r + h)

where: $r$ is the radius of the circular cross-section, $h$ is the height (or length) of the cylinder, and $\pi \approx \dfrac{22}{7}$ (or $3.14$ , whichever the numbers suggest works out more cleanly). A cylinder that is open at one or both ends — a pipe, a bucket without a lid, a drum with the top cut off — is missing one or both circular ends. In that case you cannot blindly use the TSA formula; you must reason about which surfaces actually exist and add only those.

Figure

A cylinder shown next to its unrolled net a rectangle for the curved surface and two circles for the flat ends

Derivation ATotal surface area of a cuboid from its six rectangular faces

Show that adding the areas of a cuboid's three pairs of matching rectangular faces gives the compact formula $TSA = 2(lb + bh + hl)$ , and confirm why the factor of 2 and the three products correspond exactly to the three distinct pairs of opposite faces on the solid.

Step 1. A cuboid has 6 faces, but opposite faces are congruent rectangles, so there are only 3 distinct face-shapes to compute: the base pair (top and bottom), the front-back pair, and the side pair.

Step 2. The base and top each measure length $l$ by breadth $b$ , giving area $lb$ each, so together they contribute $2lb$ .

Step 3. The front and back faces each measure breadth $b$ by height $h$ , giving area $bh$ each, so together they contribute $2bh$ .

Step 4. The two side faces each measure height $h$ by length $l$ , giving area $hl$ each, so together they contribute $2hl$ .

Step 5. Adding all three pairs gives the total surface area:

TSA = 2lb + 2bh + 2hl = 2(lb + bh + hl)

Setting $l = b = h = a$ collapses this to $TSA_{cube} = 2(a^2+a^2+a^2) = 6a^2$ , confirming the cube formula is just a special case of the cuboid one.

Worked example

A cuboidal water tank has length $2.5$ m, breadth $2$ m, and height $1.6$ m. Find its total surface area, and the cost of painting its outer surface at ₹120 per $\text{m}^2$ . Then a right circular cylindrical pipe, open at both ends, has radius $7$ cm and length $20$ cm — find its curved surface area, and the total surface area of a closed tin of the same dimensions.

Step 1 — Cuboid TSA. Using $TSA = 2(lb+bh+hl)$ with $l=2.5, b=2, h=1.6$ :

lb = 5, \quad bh = 3.2, \quad hl = 4

TSA = 2(5+3.2+4) = 2 \times 12.2 = 24.4\ \text{m}^2

Step 2 — Painting cost.

\text{Cost} = 24.4 \times 120 = ₹2928

Step 3 — Cylinder CSA. Using $CSA = 2\pi rh$ with $r=7$ cm, $h=20$ cm, $\pi = \frac{22}{7}$ :

CSA = 2 \times \frac{22}{7} \times 7 \times 20 = 44 \times 20 = 880\ \text{cm}^2

Since the pipe is open at both ends, this $880\ \text{cm}^2$ is also its full outer surface — there are no circular ends to add.

Step 4 — Closed tin TSA. A closed cylinder of the same size adds two circular ends of area $\pi r^2$ each:

2 \times \frac{22}{7} \times 7^2 = 2 \times 22 \times 7 = 308\ \text{cm}^2

TSA = 880 + 308 = 1188\ \text{cm}^2

The tank's surface area is $24.4\ \text{m}^2$ (painting cost ₹2928); the open pipe's CSA is $880\ \text{cm}^2$ ; and the closed tin's TSA is $1188\ \text{cm}^2$ .

Competitive stretch

If the same tin is instead open only at the top (a bucket-like shape), its total surface would be the curved surface plus just one circular end: $880 + 154 = 1034\ \text{cm}^2$ — always recount which ends are actually present rather than reusing a memorised formula.

Common traps

Forgetting a solid is open: adding both circular ends to a pipe, bucket, or lidless tank that is only closed on one end or none. Always check how many ends actually exist before adding $\pi r^2$ terms.
Mixing up CSA and TSA: CSA is only the curved part, $2\pi rh$ ; TSA adds the two circles, $2\pi rh + 2\pi r^2$ . Answering with TSA when a question asks only for CSA (or vice versa) is a very common slip.
Using the diameter instead of the radius: many problems state the diameter of a cylinder. Halve it to get $r$ before substituting into $2\pi rh$ or $\pi r^2$ .
Dropping the square on the unit: surface area is always in square units ( $\text{cm}^2$ , $\text{m}^2$ ) — writing the final answer in plain cm or m is an easy but wrong shortcut.

Ask the tutor: Not sure whether your cylinder problem needs CSA or TSA, or how to handle a solid that is open at one end — ask the AI tutor to work through it with you, step by step, with a fresh example.

Why do we peel the cylinder's curved surface into a rectangle instead of just measuring it directly on the round shape

A curved surface cannot be measured with an ordinary length-times-breadth formula while it stays curved — area formulas need flat shapes. Imagining the label of a tin can peeled off and laid flat turns the curved side into an honest rectangle: one side is the cylinder's height $h$ , and the other is the distance once around the circle, its circumference $2\pi r$ . Multiplying those two flat lengths gives $2\pi rh$ correctly.

If a question just says cylinder without saying open or closed, which formula should I use

Default to a closed cylinder and use the full $TSA = 2\pi r(r+h)$ , since that is the complete solid unless the question specifically describes it as a pipe, an open tank, or a bucket without a lid. Always reread the question once more for words like "open," "hollow tube," or "without a lid" before finalizing your formula, since these change which surfaces you must include.

Do I need to remember three separate formulas, or is there a shortcut

You really only need one idea: add up the areas of the faces that exist. The cuboid and cube formulas are the same idea with different numbers of distinct face-pairs, and the cylinder formula is the same idea with a curved face unrolled into a rectangle. Once you can picture a solid's net (its faces laid flat), you can rebuild any of these formulas from scratch instead of memorising three unrelated expressions.

Surface area

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads