An easy question on solids for P5, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P5 and P6 first rounds and finals: painted small cubes, water levels and the surface area of stacks.
The question
6 identical small cubes in a row (top picture) have a surface area of 104 cm².
The same 6 cubes are rebuilt into a cuboid 3 cubes long, 2 wide and 1 high (bottom picture).
What is the surface area of the cuboid, in cm²?
Show the hints (2)
💡Hint 1: Surface area is all the little squares on the outside. How many squares show on the row?
💡Hint 2: Find the area of one square, then count the squares on the outside of the cuboid.
Worked solution
Have a go first, then read the solution step by step.
1.The row of 6 cubes: the top, bottom, front and back each have 6 squares, and each end has 1 — 6 × 4 + 2 = 26 squares in all.
2.Each square's area = 104 ÷ 26 = 4 cm².
3.The cuboid: the top and bottom have 6 squares each, the front and back 3, the left and right 2 — (6 + 3 + 2) × 2 = 22 squares.
4.Surface area = 22 × 4 = 88 cm².
⚠️Common mistake: thinking the surface area stays the same because the cubes do. The volume stays the same, but the surface area depends on how many faces show: 26 in the row, only 22 on the cuboid.
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