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
3 identical cuboids are each 8 cm long, 7 cm wide and 5 cm high (shown).
They are glued face to face into one big cuboid.
What is the smallest surface area the big cuboid can have, in cm²?
Show the hints (3)
💡Hint 1: What does each join take off the surface area?
💡Hint 2: For the least surface, hide the biggest faces.
💡Hint 3: One cuboid's surface is 262 cm². 3 in a line take 2 joins, each hiding 2 of the biggest, 8 × 7, faces.
Worked solution
Have a go first, then read the solution step by step.
1.One cuboid's surface area = (8 × 7 + 8 × 5 + 7 × 5) × 2 = 262 cm², so 3 have 786 cm². Each join hides one face of each cuboid: 2 faces gone.
2.3 in a line take 2 joins. For the smallest surface, glue the biggest faces: 8 × 7 = 56 cm².
3.Surface area = 786 − 2 × 2 × 56 = 562 cm².
⚠️Common mistake: taking off one face per join. Where two cuboids are glued, both their faces are hidden.
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