Cheat sheet · Verbal Reasoning
Cube and Cuboid
Your complete guide to Cube and Cuboid problems for competitive exams. This cheat sheet helps you solve painted cube/cuboid problems and calculate volumes, surface areas, and diagonals quickly.
- Beginner
- 2 pages
- High-resolution PNG
- 202 downloads
- Added Jun 27, 2026
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Inside this cheat sheet
📦 Cube & Cuboid Basics
- Cube: Faces, Edges & Vertices
- Cuboid: Faces, Edges & Vertices
- Volume of Cube & Cuboid
- Surface Area (LSA & TSA)
- Diagonal of Cube & Cuboid
- Properties & Formulas
🎨 Painted Cube Problems
- Painting Outer Surface
- Cutting into Smaller Cubes
- Painted Cuboid Problems
- 3 Faces Painted (Corners)
- 2 Faces Painted (Edges)
- 1 Face Painted (Faces)
🔢 Cutting & Counting
- Cube: n × n × n
- Cuboid: a × b × c
- 0 Faces Painted (Inner Cubes)
- 1 Face Painted (Face Cubes)
- 2 Faces Painted (Edge Cubes)
- 3 Faces Painted (Corner Cubes)
📊 Key Formulas
- Volume = a³ (Cube) / l×b×h (Cuboid)
- TSA = 6a² (Cube)
- LSA = 4a² (Cube)
- Diagonal = a√3 (Cube)
- TSA = 2(lb+bh+lh) (Cuboid)
- Diagonal = √(l²+b²+h²) (Cuboid)
🎯 Problem Types
- Finding Painted Cubes
- Finding Unpainted Cubes
- Volume Calculations
- Surface Area Problems
- Diagonal Calculations
- Mixed Problems
📚 Exam Categories
- SSC & Banking
- Railways & UPSC
- CAT & MBA Entrance
- Defence & Insurance
- Placement Tests
- Other Competitive Exams
📦 Cube
6 Faces | 12 Edges
8 Vertices🔴 3 Faces Painted
Always 8
Corners🟢 0 Faces Painted
(n-2)³
Inner Cubes📐 Diagonal
a√3 (Cube)
√(l²+b²+h²) (Cuboid)🎨 Painted Cube - n = 5 Visualization
🔴
3 Faces
Corners
8
🟠
2 Faces
Edges
12(n-2) = 36
🟡
1 Face
Faces
6(n-2)² = 54
🟢
0 Faces
Inner
(n-2)³ = 27
📦 Cube
📦 Cuboid
🎯 Quick Problem Examples
📦 Cube Example
- Question: Side = 5 cm
- Volume: 5³ = 125 cm³
- TSA: 6×25 = 150 cm²
- Diagonal: 5√3 = 8.66 cm
- Answer: V=125, TSA=150
🎨 Painted Cube
- Question: n = 4 cube painted
- 3 Faces: 8 (corners)
- 2 Faces: 12(4-2) = 24
- 1 Face: 6(4-2)² = 24
- 0 Face: (4-2)³ = 8
- Check: 8+24+24+8=64 ✅
📦 Cuboid Example
- Question: l=4, b=3, h=2
- Volume: 4×3×2 = 24 cm³
- TSA: 2(12+6+8) = 52 cm²
- Diagonal: √(16+9+4) = √29
- Answer: V=24, TSA=52
📋 Quick Reference: Key Formulas
📦 Cube (Side = a)
- Volume: a³
- TSA: 6a²
- LSA: 4a²
- Diagonal: a√3
- Total Cubes: n³
📦 Cuboid (l × b × h)
- Volume: l × b × h
- TSA: 2(lb+bh+lh)
- LSA: 2h(l+b)
- Diagonal: √(l²+b²+h²)
- Total Cubes: a×b×c
🎨 Painted Cube (n × n × n)
- 3 Faces: 8
- 2 Faces: 12(n-2)
- 1 Face: 6(n-2)²
- 0 Faces: (n-2)³
- Total: n³
🎯 How to Solve Cube & Cuboid Problems Quickly
The Golden Rule: In painted cube problems, always subtract 2 from each dimension to find the number of inner cubes. The formula (n-2)³ gives you the number of unpainted inner cubes.
Step-by-Step Approach:
- Step 1: Identify the dimensions (n for cube, a×b×c for cuboid)
- Step 2: Determine how many faces are painted
- Step 3: Apply the appropriate formula
- Step 4: Calculate the number of small cubes with 0, 1, 2, or 3 painted faces
- Step 5: Verify your answer by checking if the sum matches total cubes
Common Mistakes to Avoid:
- ❌ Forgetting that edge cubes are counted twice when calculating 2-face painted cubes
- ❌ Confusing LSA (Lateral Surface Area) with TSA (Total Surface Area)
- ❌ Using the wrong formula for cuboid problems
- ❌ Mixing up the formulas for 0, 1, 2, and 3 painted faces
Proven Strategy: The "Subtract 2" technique – for any painted solid, subtract 2 from each dimension to find the inner layer. This inner layer has 0 painted faces. Work outward from there to find cubes with 1, 2, or 3 painted faces.
Memory Trick: Remember the Cube Formulas:
- 🔴 3 Faces Painted: Always 8 (corners)
- 🟠 2 Faces Painted: 12 × (n-2)
- 🟡 1 Face Painted: 6 × (n-2)²
- 🟢 0 Faces Painted: (n-2)³
Check: 8 + 12(n-2) + 6(n-2)² + (n-2)³ = n³ ✓
💡 Pro Tips for Cube & Cuboid Problems
✅ Know the Formulas
Memorize all cube and cuboid formulas – they are frequently asked
✅ Visualize the Cube
Picture the cube in 3D – corners, edges, and faces are key
✅ Use the Sum Check
Always verify: 8 + 12(n-2) + 6(n-2)² + (n-2)³ = n³
✅ Practice with Variants
Practice cubes, cuboids, and different painting patterns
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