Ever wondered how much wrapping paper a gift box needs?
Aisha is wrapping a birthday present for her brother. The paper must cover every face of the box, so the amount she needs is the box's surface area.
Revision notes
Add up the areas of all the faces of cubes, cuboids and cylinders — for wrapping, painting and building tanks.
Aisha is wrapping a birthday present for her brother. The paper must cover every face of the box, so the amount she needs is the box's surface area.
The surface area of a solid is the total area of all its faces. Imagine unfolding the box flat (its net) and finding the area of the whole shape.
It is an area, so it is measured in square units (cm², m²) — not cubic units.
🔑 Key idea: Surface area = add up the areas of every face.
A cube has 6 equal square faces: surface area = 6a². A cube with 3 cm edges has 6 × 9 = 54 cm².
A cuboid has 3 pairs of equal rectangles: top and bottom (l × w), front and back (l × h), and two ends (w × h). So surface area = 2(lw + lh + wh).
For a box 5 cm by 4 cm by 2 cm: 2 × (20 + 10 + 8) = 2 × 38 = 76 cm².
An open box or tank has no lid, so leave out the top: lw + 2lh + 2wh.
🔑 Key idea: Cube: 6a². Cuboid: 2(lw + lh + wh). Open box: drop the lid.
Unroll the curved side of a cylinder and you get a rectangle. Its length is the circumference, 2πr, and its width is the height h, so the curved surface area = 2πrh.
A closed cylinder also has two circles, each πr². Total surface area = 2πrh + 2πr² = 2πr(r + h). We usually take π = 227.
🔑 Key idea: Cylinder: curved = 2πrh; total = 2πr(r + h).
A closed box is 10 cm long, 4 cm wide and 7 cm high. Find its total surface area.
Step 1: One of each pair
lw = 10 × 4 = 40
lh = 10 × 7 = 70
wh = 4 × 7 = 28
Step 2: Add and double
2 × (40 + 70 + 28) = 2 × 138 = 276 cm²
Answer: 276 cm²
Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.