A hard question on solids for P6, 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
From a wooden cube with edges of 5 cm, a small cube with edges of 1 cm is dug out at each of these places:
3 corners, the middle of 1 edge, the middle of the top face (shown).
What is the surface area of what's left, in cm²?
Show the hints (3)
💡Hint 1: Digging out a small cube takes some small squares away, but shows some new ones.
💡Hint 2: Take each kind in turn: dug at a corner, the middle of an edge, the middle of a face — how many squares go, and how many appear?
💡Hint 3: At a corner: 3 go, 3 appear; in the middle of an edge: 2 go, 4 appear; in the middle of a face: 1 goes, 5 appear. Each small square is 1 × 1 = 1 cm².
Worked solution
Have a go first, then read the solution step by step.
1.The surface area to start with = 5 × 5 × 6 = 150 cm².
2.Dug from a corner: the 3 small squares that were on the outside go, but the notch shows 3 new ones — no change.
3.Dug from the middle of an edge: 2 small squares go, 4 appear — 2 more. Each is 1 × 1 = 1 cm², so 1 of them add 1 × 2 × 1 = 2 cm².
4.Dug from the middle of a face: 1 small square goes, 5 appear (four walls and a floor) — 4 more: 4 × 1 = 4 cm².
5.The surface area left = 150 + 2 + 4 = 156 cm².
⚠️Common mistake: thinking that digging wood out makes the surface area smaller. Digging shows new faces: at a corner nothing changes; on an edge or a face, the area grows.
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