Six rectangles laid flat
Show on a big screenWrap a carton, and the paper must cover every face. A carton is a cuboid, with six rectangle faces.
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- A cuboid has length l, breadth b, height h.
- Cut along some edges and open it out flat.
- Six rectangles: two of each size.
- Add all six for the total surface.
The rule
Cut a carton along some edges and it opens out flat. An edge is where two faces meet. That flat shape is its net.
A cuboid's net is six rectangles, in three matching pairs. Add their areas. The total is the surface area.
Let l be the length, b the breadth and h the height. Each face has a twin, so each area counts twice.
surface area = 2 × (l × b + b × h + h × l)
A carton is 8 cm long, 5 cm broad, 3 cm high. One face of each pair: 8 × 5 = 40 cm². 8 × 3 = 24 cm². 5 × 3 = 15 cm². The unit cm² means square centimetres.
| Face pair | Sides | Area of the pair |
|---|---|---|
| Top, bottom | 8 and 5 | 2 × 40 = 80 |
| Front, back | 8 and 3 | 2 × 24 = 48 |
| Two ends | 5 and 3 | 2 × 15 = 30 |
Add the pairs: 80 + 48 + 30 = 158. The carton needs 158 cm² of paper.
A cube is a cuboid with l = b = h. All six faces match. Its surface area is 6 × edge × edge. An edge of 5 cm gives 5 × 5 = 25. Then 6 × 25 = 150 cm².
Watch out!
8 × 5 × 3 = 120 is not the paper. That number counts the 1 cm cubes that fill the carton. The paper is 158 cm².
Try the lab to finish this lesson.