1
Nets and the surface area of a cuboid
Discover Mathematics P6: Term 2, Unit 13, Lessons 1–2
GoalUse a net to find the area of each face, and find the surface area of a cuboid.
New words
net · a flat pattern that folds up into a solidface · one flat side of a solidsurface area · the total area of all the faces of a solid
Explanation
A cuboid has six rectangular faces, in three equal pairs: top and bottom, front and back, left and right. If we unfold it we get a net, a flat shape with all six faces. The surface area is the sum of the areas of the six faces, measured in square units (cm² or m²). For a cuboid with length l, width w and height h, the top and bottom are l × w each, the front and back are l × h each, and the two sides are w × h each. So surface area = 2 × (l × w + l × h + w × h). In a drawing you see only three faces, so remember the three hidden faces that are equal to them.
ƒFormula
Surface area of a cuboid = 2 × (l × w + l × h + w × h)
Three pairs of equal faces.
Worked examples
A cuboid is 5 cm long, 3 cm wide and 2 cm high. The net shows the six faces: 15 + 15 + 10 + 10 + 6 + 6 = 62. The surface area is 62 cm².
With the formula: 2 × (5 × 3 + 5 × 2 + 3 × 2) = 2 × (15 + 10 + 6) = 2 × 31 = 62 cm². Same answer.
A box is 8 cm by 5 cm by 3 cm. Top and bottom: 2 × 40 = 80. Front and back: 2 × 24 = 48. Two sides: 2 × 15 = 30. Total 80 + 48 + 30 = 158 cm².
Class activity
Unfold a box. With a grown-up, carefully open a small cereal or medicine box along its edges and lay it flat. Measure the faces and find its surface area.
Practice
1
Find the surface area of a cuboid 7 cm by 3 cm by 2 cm.
2 × (21 + 14 + 6) = 2 × 41 = 82 cm².
2
Find the surface area of a cuboid 10 cm by 10 cm by 5 cm.
2 × (100 + 50 + 50) = 400 cm².
3
A carton is 20 cm by 15 cm by 10 cm. How much paper covers all six faces?
2 × (300 + 200 + 150) = 1,300 cm².
4
An open box (no lid) is 5 cm by 4 cm by 3 cm. Find the area of its five faces.
Bottom 20. Two long sides 2 × 15 = 30. Two short sides 2 × 12 = 24. Total 74 cm².
5
Why do we say the surface area is in cm² and not in cm³?
We add areas of flat faces, which are square units. Cubic units are for volume.