Unit 5 · Geometry
Painted Cuboid — How Many Unpainted Cubes?
Lesson 2 of 4
Painted-cube questions with zero red faces are always the interior core: subtract one layer from each dimension, then multiply.
Unpainted unit-cube counts appear in the medium band of Selective MR — the (n − 2) formula works for cubes and cuboids once you identify interior dimensions.
The examiner tests whether you know only fully enclosed unit cubes escape the paint, and whether you can apply (L − 2)(W − 2)(H − 2) after cutting a cuboid into 1 cm cubes.
A cuboid is painted on all faces, then cut into 1 cm unit cubes. You count how many small cubes have no painted face.
Best approach: Interior length = L − 2, same for width and height (when each is at least 2). Multiply the three interior dimensions for the count of unpainted cubes.
Question
A cuboid is 5 centimetres long, 4 centimetres wide and 3 centimetres high. Paint its six faces red and then cut it into small cubes with edge length 1 centimetre. How many small cubes have no face painted red?
Select an answer to see the explanation.
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