Cube and Cuboid— Concepts, Formulas & Shortcuts
- A cube cut into n slices per edge yields n³ small cubes.
- Exactly 3 faces painted → the 8 corner cubes, always 8 regardless of n.
- Exactly 2 faces painted → 12(n − 2), the non-corner edge cubes.
- Exactly 1 face painted → 6(n − 2)², the interior of each face.
- No face painted → (n − 2)³, the hidden core.
- Check: 8 + 12(n − 2) + 6(n − 2)² + (n − 2)³ = n³.
Cube and Cuboid Practice Questions with Answers
Attempt each question first, then open the explanation. All 10 questions below are free to read and require no signup.
Q1.A cube is painted on all faces and cut into 27 equal smaller cubes. How many have exactly 3 faces painted?
Easy- A4
- B6
- C8
- D12
Cube and Cuboid question 1 of 10+Show Answer & Explanation
Answer: C. 8
Explanation: The corner cubes always number 8, whatever the cut.
Q2.In the same 27-cube case, how many have exactly 2 faces painted?
Moderate- A8
- B10
- C12
- D16
Cube and Cuboid question 2 of 10+Show Answer & Explanation
Answer: C. 12
Explanation: 12(n − 2) with n = 3 gives 12 × 1 = 12 edge cubes.
Q3.In the same 27-cube case, how many have exactly 1 face painted?
Moderate- A4
- B6
- C8
- D12
Cube and Cuboid question 3 of 10+Show Answer & Explanation
Answer: B. 6
Explanation: 6(n − 2)² = 6 × 1 = 6 — the centre cube of each face.
Q4.In the same 27-cube case, how many have NO face painted?
Moderate- A0
- B1
- C3
- D6
Cube and Cuboid question 4 of 10+Show Answer & Explanation
Answer: B. 1
Explanation: (n − 2)³ = 1³ = 1, the single hidden core cube.
Q5.A painted cube is cut into 64 smaller cubes. How many have exactly 2 faces painted?
Moderate- A12
- B18
- C24
- D36
Cube and Cuboid question 5 of 10+Show Answer & Explanation
Answer: C. 24
Explanation: n = 4, so 12(n − 2) = 12 × 2 = 24.
Q6.In the same 64-cube case, how many have exactly 1 face painted?
Moderate- A12
- B18
- C24
- D30
Cube and Cuboid question 6 of 10+Show Answer & Explanation
Answer: C. 24
Explanation: 6(n − 2)² = 6 × 4 = 24.
Q7.In the same 64-cube case, how many have no paint at all?
Moderate- A4
- B6
- C8
- D12
Cube and Cuboid question 7 of 10+Show Answer & Explanation
Answer: C. 8
Explanation: (n − 2)³ = 2³ = 8 interior cubes.
Q8.A cube is cut into 125 smaller cubes. How many lie completely inside, unpainted?
Difficult- A9
- B18
- C27
- D36
Cube and Cuboid question 8 of 10+Show Answer & Explanation
Answer: C. 27
Explanation: n = 5, so (5 − 2)³ = 27.
Q9.How many total faces does a cube have?
Easy- A4
- B6
- C8
- D12
Cube and Cuboid question 9 of 10+Show Answer & Explanation
Answer: B. 6
Explanation: A cube has 6 square faces, 12 edges and 8 vertices.
Q10.For a painted cube cut into n³ pieces, the number with exactly three painted faces is:
Moderate- A6
- B8
- C12(n − 2)
- D(n − 2)³
Cube and Cuboid question 10 of 10+Show Answer & Explanation
Answer: B. 8
Explanation: Only the 8 corners touch three outer faces, and that count is independent of n.
Cube and Cuboid — Frequently Asked Questions
Why are there always exactly 8 cubes with three painted faces?+
Because only the corners of the large cube touch three outer surfaces, and a cube has exactly 8 corners no matter how finely it is cut.
How do I find the unpainted cubes?+
They form the inner cube left after removing one layer from every side, so the count is (n − 2)³. For a cube cut into 4 per edge, that is 2³ = 8.
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