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
A cuboid's top, front and right faces have areas of 35 cm², 20 cm² and 28 cm² (shown); its length, width and height are whole numbers of cm.
What is its volume, in cm³?
Show the hints (2)
💡Hint 1: Top = length × width, front = length × height, right = width × height. Which faces share an edge?
💡Hint 2: Top × front, divided by right: what's left?
Worked solution
Have a go first, then read the solution step by step.
1.The top (length × width) times the front (length × height), divided by the right (width × height): the width and height cancel. 35 × 20 ÷ 28 = 25, so length × length = 25.
2.So the length = 5 cm. Width = 35 ÷ 5 = 7 cm, height = 20 ÷ 5 = 4 cm.
3.Volume = 5 × 7 × 4 = 140 cm³.
⚠️Common mistake: multiplying the three areas: 35 × 20 × 28 = 19600. That's (length × width × height) × (length × width × height) — the volume times itself.
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