Surface area of a cube and a cuboid

Surface area as the area of all faces: a net of a solid and the formula 2 × (ab + bc + ac).

How much paper to wrap a box

You wrap a present. How much wrapping paper do you need? Exactly as much as the surface area of the box — the sum of the areas of all its faces.

A shoebox is a cuboid: a solid with 6 faces, all rectangles. A dice is a cube: also 6 faces, but all the same squares.

The best trick for surface area: unfold the solid into a flat shape. Cut the box along the edges and lay it on the table — you get a pancake of six rectangles, called a net of the solid. And the area of a pancake you already can count: the area of each rectangle with the formula S = a × b, then add.

With a cuboid you shorten the work by looking: the faces come in pairs. The bottom and the lid are the same, front and back the same, left and right the same. Three different rectangles, each twice — that is why the formula S = 2 × (a·b + b·c + a·c).

Surface area is still just area — only gathered from every side of the box. New topic? More like old topic six times.

Sofie wraps mum a present — a box 30 cm long, 20 cm wide and 10 cm high. She estimates the paper by eye, cuts, and a strip is missing over a corner. Jonáš, who is watching, takes paper and a pencil: “We’ll count it. Bottom and lid: twice 30 times 20 is 1 200. Front and back: twice 30 times 10 is 600. Sides: twice 20 times 10 is 400.” He adds: 2 200 cm². “Buy a sheet 50 by 50, that is 2 500 — and there will be leftover for folds.” Second try: the paper is enough, the present wrapped. The estimate failed, the calculation did not.

💡

Always sketch the box and label the edges a, b, c. Then you see three pairs of faces in front of you: ab (bottom+lid), bc (sides), ac (front+back).

A net, pairs of faces and formulas

Method through pairs of faces (a cuboid with edges a, b, c):

Bottom and lid: rectangles a × b. Together 2 × a × b.
Front and back face: rectangles a × c. Together 2 × a × c.
Left and right side: rectangles b × c. Together 2 × b × c.
Add: S = 2·ab + 2·ac + 2·bc, shortened S = 2 × (ab + ac + bc).

In numbers for a box 30 × 20 × 10: ab = 600, ac = 300, bc = 200. Sum 1 100, times two = 2 200 cm².

A cube has all 6 faces the same — squares a × a:

S = 6 × a × a. A cube with edge 4 cm: S = 6 × 16 = 96 cm².

Checks that will hold you:

Surface area is a surface → unit cm² (m²). If cm³ comes out, you mixed surface area with volume (that comes next).
In the brackets belong three different products — each pair of edges exactly once: ab, ac, bc. Who writes ab twice counts one face four times and another not at all.
A net estimate: six faces of a shoebox ≈ six A5 papers ≈ three A4 papers. Order of thousands of cm², not tens.

Surface area of a cuboid 5 × 4 × 3 cm

Three pairs of faces:

ab = 5 × 4 = 20 cm²
ac = 5 × 3 = 15 cm²
bc = 4 × 3 = 12 cm²

Sum of the brackets: 20 + 15 + 12 = 47 cm².

Surface area: S = 2 × 47 = 94 cm².

Check: six faces, none bigger than 20 cm² → the surface area must be under 6 × 20 = 120 cm² and over 6 × 12 = 72 cm². Ninety-four sits in the fork.

Surface area of a cube with edge 6 cm

A cube = 6 identical square faces.

One face: 6 × 6 = 36 cm².

Surface area: S = 6 × 36 = 216 cm².

A fun fact for players: a classic dice has an edge of about 1.6 cm — surface area 6 × 2.56 ≈ 15 cm². A Rubik’s cube with edge 5.7 cm has surface area almost 195 cm². The same shape, thirteen times more area to sticker.

Painting a chest: surface area without a lid

A wooden toy chest (60 × 40 × 30 cm, no lid!) is to be painted on the outside. How many cm² are painted?

The whole surface area: ab = 2 400, ac = 1 800, bc = 1 200 → S = 2 × 5 400 = 10 800 cm².

But the lid is missing — I subtract one face a × b: 10 800 − 2 400 = 8 400 cm².

The lesson: the formula is a start, not an end. A tank has no lid, a room is painted without the floor — always think WHICH faces really count.

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Surface area is not volume. Surface area = the area of the faces from the outside (paper for wrapping, cm²). Volume = the space inside (what fits inside, cm³). The word problem hints: “paint, wrap, sticker” = surface area; “pour, fill, fits in” = volume.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Work out the surface area of a cuboid: S = 2 × (ab + bc + ac). Write the three products on paper. The answer is a number (in cm²).

Practise the questions

On paper

First cut, then count — a net of a solid once in your hands is worth ten formulas.

Take a small box (from tea, from matches), cut it along the edges and unfold it into a net. Measure the faces with a ruler, work out the area of each and add the surface area.
Work out the surface area: a cuboid 8 × 5 × 2 cm and a cube with edge 7 cm. For the cuboid list all three pairs of faces.
Draw a net of a cube with edge 3 cm on squared paper (six squares in a cross shape), cut it out and glue a cube.
A tank 50 × 30 × 30 cm has no lid. How many cm² of glass is needed? (Watch: you count the bottom and four sides.)

Now you 💪

I can name three pairs of faces of a cuboid and their sizes (ab, ac, bc).
I can work out the surface area of a cuboid with the formula and of a cube through 6 identical squares.
From the question I can tell whether it asks for surface area (wrapping, painting), or volume (filling).

✅

Done when: You work out the surface area of a cube and a cuboid through pairs of faces, you unfold a solid into a net, and you can leave out faces that do not count.

What to take from this lesson

Surface area = the sum of the areas of all faces. Square units (cm²).
Cuboid: three pairs of faces → S = 2 × (ab + ac + bc).
Cube: six identical squares → S = 6 × a × a.
A net of a solid = a box unfolded into a pancake. The best path to understanding surface area.
Read the question: wrap and paint = surface area, pour and fill = volume. And sometimes a face is missing!