Cube and Cuboid Questions for Placement 2026 (with Solutions)

Last Updated: March 2026

Introduction to Cube and Cuboid

Cube and Cuboid problems are a specialized topic in spatial reasoning and mensuration that appear frequently in placement exams. These questions test your ability to visualize 3D objects, understand surface area and volume relationships, and solve problems related to cutting, painting, and rearranging cubes. This topic is particularly important for TCS, Infosys, and Wipro exams.

Why This Topic is Important

Cube and Cuboid questions assess:

3D spatial visualization ability
Understanding of surface area and volume
Pattern recognition with painted cubes
Logical reasoning with cut pieces
Quick calculation with geometric formulas

Companies That Ask Cube and Cuboid Questions (with Frequency)

Company	Frequency	Difficulty Level
TCS	Very High	Moderate to High
Infosys	High	Moderate
Wipro	Moderate	Easy to Moderate
Cognizant	Moderate	Moderate
Accenture	Moderate	Easy
Capgemini	Low	Moderate
IBM	Low	Moderate
Tech Mahindra	Moderate	Easy to Moderate
HCL	Low	Moderate
LTI Mindtree	Low	Moderate

KEY FORMULAS / CONCEPTS

╔══════════════════════════════════════════════════════════════════╗
║                 CUBE AND CUBOID FORMULA SHEET                    ║
╠══════════════════════════════════════════════════════════════════╣
║                                                                  ║
║  CUBE (Side = a)                                                 ║
║  ────────────────────────────────────────────────────────────   ║
║  Volume = a³                                                     ║
║  Total Surface Area (TSA) = 6a²                                 ║
║  Lateral Surface Area (LSA) = 4a²                               ║
║  Face Diagonal = a√2                                            ║
║  Space Diagonal = a√3                                           ║
║                                                                  ║
║  CUBOID (l × b × h)                                              ║
║  ────────────────────────────────────────────────────────────   ║
║  Volume = l × b × h                                             ║
║  Total Surface Area = 2(lb + bh + hl)                           ║
║  Lateral Surface Area = 2h(l + b)                               ║
║  Space Diagonal = √(l² + b² + h²)                               ║
║                                                                  ║
║  PAINTED CUBE FORMULAS (n × n × n cube cut into n³ small cubes) ║
║  ────────────────────────────────────────────────────────────   ║
║                                                                  ║
║  3 faces painted (corners): Always = 8                          ║
║                                                                  ║
║  2 faces painted (edges): = 12(n - 2)                           ║
║                                                                  ║
║  1 face painted (face centers): = 6(n - 2)²                     ║
║                                                                  ║
║  0 faces painted (interior): = (n - 2)³                         ║
║                                                                  ║
║  Total small cubes: n³ = 8 + 12(n-2) + 6(n-2)² + (n-2)³        ║
║                                                                  ║
║  CUBOID CUTTING (a × b × c pieces)                              ║
║  ────────────────────────────────────────────────────────────   ║
║  3 faces painted: 8 (always)                                    ║
║  2 faces painted: 4[(a-2) + (b-2) + (c-2)]                      ║
║  1 face painted: 2[(a-2)(b-2) + (b-2)(c-2) + (c-2)(a-2)]       ║
║  0 faces painted: (a-2)(b-2)(c-2)                               ║
║                                                                  ║
╚══════════════════════════════════════════════════════════════════╝

30 Practice Questions with Step-by-Step Solutions

Question 1

A cube of side 6 cm is painted red on all faces and cut into 1 cm cubes. How many small cubes have exactly 2 faces painted?

Solution: n = 6/1 = 6 2 faces painted = 12(n-2) = 12 × 4 = 48

Question 2

A 5×5×5 cube is painted and cut into 1×1×1 cubes. How many cubes have no face painted?

Solution: 0 faces painted = (n-2)³ = (5-2)³ = 3³ = 27

Question 3

A cube is painted blue on opposite faces, red on two adjacent faces, and green on remaining faces. It is cut into 64 small cubes. How many have at least one face painted red?

Solution: n = 4 (since 4³ = 64) Total cubes = 64 Cubes with no red = cubes from non-red faces Red faces are 2 adjacent faces, so 4 faces have no red The unpainted interior + green/blue only regions Easier: Total - (cubes with no paint at all or only green/blue) Actually: 2 faces red, each has (n-2)² = 4 edge cubes shared Red only cubes = 2×(n-2)² - edges + corners Simple way: Each red face has 16 small faces. Adjacent edge shares 4 cubes with both red. Total with red = 2×16 - 4 = 28

Question 4

A cuboid of dimensions 8×6×4 is painted and cut into 1×1×1 cubes. How many cubes have exactly 1 face painted?

Solution: 1 face painted = 2[(a-2)(b-2) + (b-2)(c-2) + (c-2)(a-2)] = 2[6×4 + 4×2 + 2×6] = 2[24 + 8 + 12] = 2 × 44 = 88

Question 5

A cube is cut into 343 identical small cubes. How many small cubes have exactly one face painted?

Solution: n³ = 343 → n = 7 1 face painted = 6(n-2)² = 6 × 25 = 150

Question 6

A 10×8×6 cuboid is painted on all surfaces and cut into 1×1×1 cubes. How many cubes have paint on exactly 3 faces?

Solution: 3 faces painted = 8 (always, at corners)

Question 7

A cube of side 4 cm is cut into 64 equal small cubes. What is the ratio of total surface area of all small cubes to the surface area of original cube?

Solution: Original TSA = 6 × 16 = 96 cm² Each small cube: side = 0.5 cm, TSA = 6 × 0.25 = 1.5 cm² Total of 64 cubes = 64 × 1.5 = 96 cm² Wait, recheck: 4 cm cut into 64 = 4×4×4, so each small cube is 1×1×1 Small cube TSA = 6 × 1 = 6 cm² 64 cubes: 64 × 6 = 384 cm² Original: 6 × 16 = 96 cm² Ratio = 384:96 = 4:1

Question 8

A cube painted on all faces is cut into 125 small cubes. How many have at least two faces painted?

Solution: n = 5 (since 5³ = 125) 3 faces: 8 2 faces: 12(n-2) = 12 × 3 = 36 At least 2 faces = 8 + 36 = 44

Question 9

A wooden cube with edge 6 cm is cut into 216 cubes of 1 cm edge. How many cubes have exactly two faces painted if only opposite faces were painted?

Solution: Only 2 opposite faces painted (not all 6) On each painted face: edge cubes have 2 painted faces, but corners would have only 1 face painted in this case (only one pair of opposite faces painted) Actually: 2 opposite faces painted means only the edges where these faces meet other faces have relevant painting 2-face painted cubes appear only at the edges connecting the two painted faces... but painted faces are opposite, so they don't share edges! So 2-face painted cubes = 0

Question 10

A 7×5×3 cuboid is painted and cut. How many small cubes have no face painted?

Solution: 0 faces painted = (a-2)(b-2)(c-2) = (7-2)(5-2)(3-2) = 5 × 3 × 1 = 15

Question 11

A cube of side 9 cm is cut into 3 cm cubes. How many small cubes are there and how many have exactly two faces painted?

Solution: n = 9/3 = 3 Total cubes = 3³ = 27 2 faces painted = 12(n-2) = 12 × 1 = 12

Question 12

64 small cubes of 1 cm edge are arranged to form a large cube. The large cube is painted yellow on all faces and then separated. How many small cubes have no yellow face?

Solution: n = 4 (since 4³ = 64) 0 faces painted = (4-2)³ = 2³ = 8

Question 13

A cube is cut into 512 small cubes. How many have exactly one face painted?

Solution: n³ = 512 → n = 8 1 face painted = 6(n-2)² = 6 × 36 = 216

Question 14

A 6×4×3 cuboid is painted on all faces and cut. Find the number of cubes with paint on exactly two faces.

Solution: 2 faces painted = 4[(a-2) + (b-2) + (c-2)] = 4[4 + 2 + 1] = 4 × 7 = 28

Question 15

A solid cube is cut into 27 identical small cubes. What fraction of total surface area of small cubes is not painted if the original cube was painted on all faces?

Solution: n = 3 Original TSA = 6 × (side)² Let original side = 3, small cube side = 1 Original TSA = 6 × 9 = 54 Total TSA of 27 small cubes = 27 × 6 = 162 Painted surface on small cubes = Original TSA = 54 Unpainted = 162 - 54 = 108 Fraction = 108/162 = 2/3

Question 16

A 12 cm cube is painted red and cut into 2 cm cubes. How many small cubes have at most one face painted?

Solution: n = 12/2 = 6 0 faces = (6-2)³ = 64 1 face = 6(6-2)² = 6 × 16 = 96 At most 1 face = 64 + 96 = 160

Question 17

A cuboid 9×7×5 is painted and cut into 1×1×1 cubes. How many cubes have paint on exactly one face?

Solution: 1 face painted = 2[(9-2)(7-2) + (7-2)(5-2) + (5-2)(9-2)] = 2[7×5 + 5×3 + 3×7] = 2[35 + 15 + 21] = 2 × 71 = 142

Question 18

A cube is painted such that three faces meeting at a corner are red and the other three are blue. It is cut into 64 small cubes. How many have both red and blue paint?

Solution: n = 4 Cubes with both colors are along the edges where red and blue faces meet There are 3 edges where red meets blue (at the corner where 3 red faces meet, opposite corner has 3 blue) Each such edge has (n-2) = 2 cubes with both colors Plus the corners themselves (3 corners with mixed? No, corners are pure) Actually: The edges connecting red faces to blue faces Each of the 3 edges from the "red corner" to the "blue corner" direction Total edges = 3 × (n-2) + ... corners? At edges: (n-2) = 2 cubes per edge, 3 such edges = 6 Plus 3 corners where exactly 2 faces meet (1 red + 1 blue)? No corner has mixed colors. Answer: 6 small cubes have both colors

Question 19

An 8 cm cube is cut into 2 cm cubes. How many of the smaller cubes have exactly two painted faces?

Solution: n = 8/2 = 4 2 faces painted = 12(n-2) = 12 × 2 = 24

Question 20

A 5×4×3 cuboid is cut into 60 unit cubes and painted after cutting. How many cubes have exactly two faces painted?

Solution: If painted after cutting, each cube is separate and painted individually Each unit cube has 6 faces painted! But this seems like a trick question. Assuming standard "paint then cut": 2 faces = 4[(5-2) + (4-2) + (3-2)] = 4[3+2+1] = 24

Question 21

A cube is cut into 1000 small cubes. How many have paint on exactly three faces?

Solution: 3 faces painted = 8 (always, corners only)

Question 22

A wooden block 15×12×9 is cut into 1×1×1 cubes. How many cubes have no paint if the block was painted before cutting?

Solution: 0 faces painted = (15-2)(12-2)(9-2) = 13 × 10 × 7 = 910

Question 23

A cube of side 6 is painted and cut. How many small cubes (1×1×1) have at least one face painted?

Solution: n = 6 Total cubes = 216 0 faces = (6-2)³ = 64 At least 1 face = 216 - 64 = 152

Question 24

A 10×8×6 cuboid painted on all faces is cut into 1×1×1 cubes. How many cubes have paint on exactly two faces?

Solution: 2 faces painted = 4[(10-2) + (8-2) + (6-2)] = 4[8 + 6 + 4] = 4 × 18 = 72

Question 25

A cube is divided into 216 small identical cubes. How many small cubes have exactly one face painted if only four faces of the original cube were painted?

Solution: n = 6 (since 6³ = 216) If only 4 faces painted (say, leaving top and bottom): Each painted face has (n-2)² = 16 cubes with exactly 1 face painted But edges are shared... actually 1-face painted cubes on each face exclude edges = 4 × (n-2)² = 4 × 16 = 64 (If the 4 faces are adjacent, need adjustment, but standard is 64)

Question 26

A solid metal cube of side 12 cm is melted and recast into 8 smaller cubes of equal size. What is the side of each small cube and what is the ratio of surface areas?

Solution: Volume of large cube = 12³ = 1728 cm³ Volume of each small cube = 1728/8 = 216 cm³ Side of small cube = ∛216 = 6 cm TSA of large cube = 6 × 144 = 864 cm² TSA of 8 small cubes = 8 × 6 × 36 = 1728 cm² Ratio = 864:1728 = 1:2

Question 27

A 9×6×3 cuboid is painted and cut into 1×1×1 cubes. Find total number of painted faces on all small cubes.

Solution: Total TSA of cuboid = 2(9×6 + 6×3 + 3×9) = 2(54 + 18 + 27) = 2 × 99 = 198 Each painted small face on the cuboid becomes a painted face on a small cube Total painted faces on small cubes = 198

Question 28

A cube is cut into 729 small cubes. How many have exactly two faces painted?

Solution: n³ = 729 → n = 9 2 faces painted = 12(n-2) = 12 × 7 = 84

Question 29

A 7×5×4 cuboid is painted and cut. How many cubes have exactly 3 faces painted?

Solution: 3 faces painted = 8 (always at corners)

Question 30

A cube of side 5 cm is cut into 1 cm cubes. If the original cube was painted on two opposite faces only, how many small cubes have exactly one face painted?

Solution: Only 2 opposite faces painted Each painted face has (n-2)² = 9 cubes with exactly 1 face painted Total = 2 × 9 = 18

SHORTCUTS & TRICKS

Trick 1: Painted Cube Formulas

For n×n×n cube cut into 1×1×1:

3 faces: Always 8 (corners)
2 faces: 12(n-2) (edges)
1 face: 6(n-2)² (face centers)
0 faces: (n-2)³ (interior)

Trick 2: Quick n Calculation

If cube of side S is cut into s×s×s small cubes: n = S/s (number of divisions per edge)

Trick 3: Total Cube Count Verification

Always verify: 8 + 12(n-2) + 6(n-2)² + (n-2)³ = n³

Trick 4: Cuboid Variations

For a×b×c cuboid:

Memorize the pattern: 4[(a-2)+(b-2)+(c-2)] for 2-face
2[(a-2)(b-2)+(b-2)(c-2)+(c-2)(a-2)] for 1-face

Trick 5: Partial Painting

If only some faces painted:

Calculate per painted face: (n-2)² for 1-face cubes
Adjust for shared edges

Trick 6: Volume Conservation

When recasting: Volume before = Volume after Use this to find unknown dimensions.

Trick 7: Surface Area Ratio

When cube cut into n³ pieces: Ratio of total small TSA to original TSA = n

Common Mistakes to Avoid

Confusing n with side length: n is the number of divisions, not the side length of original cube.
Edge Count Error: Remember there are 12 edges in a cube, not 8 or 6.
Partial Painting: Read carefully whether all faces or only some faces are painted.
Cuboid vs Cube Formulas: Don't use cube formulas for cuboids - the cuboid formulas are different!
"At least" vs "Exactly": "At least 2 faces" means 2 or 3 faces, not just 2.
Interior Cubes: For (n-2)³, remember n must be > 2 for any interior cubes.
Unit Consistency: Ensure all dimensions are in the same units before calculating.

5 Frequently Asked Questions

Q1: Do I need to memorize all the formulas? A: Yes! The painted cube formulas are standard and appear frequently. Practice deriving them to understand better.

Q2: What if the cuboid dimensions are different? A: Use the cuboid-specific formulas. Don't try to force cube formulas onto cuboid problems.

Q3: How do I handle partially painted cubes? A: Count painted faces on each small cube based on its position (corner, edge, face center, interior).

Q4: Are cube problems always about painted surfaces? A: No, they can also involve volume, surface area, melting and recasting, or spatial arrangement.

Q5: What's the best way to visualize these problems? A: Draw diagrams! Even rough sketches help. For cubes, imagine a Rubik's cube and identify piece types.

Master these 30 questions and you'll be well-prepared for Cube and Cuboid questions in your placement exams!