Surface Area

Volume measures how much fits inside; surface area measures how much it takes to cover outside. Wrapping paper for a gift box, a coat of paint for a tin, wallpaper for a room — all of them ask for the total area of every face of a solid.

Surface area: unfold the solid

Surface area is the sum of the areas of all the faces of a solid. The easiest strategy is to unfold: imagine slitting along the edges and flattening out, and a 3D problem instantly becomes a 2D area problem — count the faces, measure each one, add them up.

The cube is simplest: 6 identical faces, each $a^2$ , so

$S = 6a^2$

A cube of edge 3 cm: $6 \times 3^2 = 54$ — wrapping it takes 54 cm² of paper.

InteractiveUnfold the Surface

Length l4Width w3Height h2

V = l × w × h = 4 × 3 × 2 = 24

Bottom layer

Layers

Drag the edge length, but keep your eye on the area of one face, $a^2$ , then multiply by 6 in your head — the surface area is six identical faces stitched together.

The cuboid: three pairs of twins

A cuboid's 6 faces come in matching pairs: top-bottom, front-back, left-right. For length $l$ , width $w$ , height $h$ :

$S = 2(lw + lh + wh)$

Walk through a 4 × 3 × 2 box:

$S = 2 \times (4 \times 3 + 4 \times 2 + 3 \times 2) = 2 \times (12 + 8 + 6) = 52 \ \mathrm{cm^2}$

Measure the three different faces first, then double to pay for their twins — much faster than counting all six one by one.

The cylinder: two discs and a rolled-up sheet

A cylinder's side wall is a rolled-up rectangle: unrolled, it is $2\pi r$ long (the circumference of the base) and $h$ tall; add the two circular discs, each $\pi r^2$ . So

$S = 2\pi r^2 + 2\pi r h$

Tins, cups, oil drums — every side wall that was "rolled" answers to this rectangle.

Covering it, or filling it

Surface area and volume are the easiest pair to confuse; one sentence sorts them out:

Surface area — painted on the outside: paint, wrapping paper, sheet metal; units are square (cm²);
Volume — poured into the inside: water, sand, air; units are cubic (cm³).

The wrapping-paper problem

A gift box measures 30 × 20 × 10 cm. How much wrapping paper at minimum?

$S = 2 \times (30 \times 20 + 30 \times 10 + 20 \times 10) = 2 \times (600 + 300 + 200) = 2200 \ \mathrm{cm^2}$

Two thousand two hundred square centimetres — about four sheets of A4 (roughly 600 cm² each). Real wrapping needs a little extra for seams and folds.

Don't skip the hidden face

A box on the table shows only 5 faces; the bottom, though hidden, still counts. Surface area covers all the faces — unless the problem explicitly says "ignore the base".

Check yourself

Quick quiz

1. What is the surface area of a cube with edge 3 cm?
2. What is the surface area of a 4 × 3 × 2 cuboid?
3. Unrolled flat, what shape is a cylinder's side wall?