A medium question on solids for P5–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
A wooden bar is a cuboid with square ends.
It is sawn into 4 equal cubes (shown), and the total surface area goes up by 24 cm².
What was the bar's volume, in cm³?
Show the hints (2)
💡Hint 1: How many cuts make 4 pieces?
💡Hint 2: Each cut makes how many new faces? What shape is each one?
Worked solution
Have a go first, then read the solution step by step.
1.Cutting into 4 pieces takes 3 cuts.
2.Each cut adds a square on each side of it: 2 faces. That's 3 × 2 = 6 new squares, each 24 ÷ 6 = 4 = 2 × 2 cm², so the square's side is 2 cm.
3.The bar is 4 × 2 = 8 cm long: volume = 8 × 2 × 2 = 32 cm³.
⚠️Common mistake: thinking 4 pieces take 4 cuts, or that each cut adds only one face.
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