Cube and Cuboid
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Cheat Sheets
Study MaterialCube and Cuboid – Cheat Sheet
Cube and Cuboid problems are frequently tested in Quantitative Aptitude and Reasoning sections of SSC, Banking, Railway, UPSC, Defence, Insurance, CAT, and other competitive examinations.
This cheat sheet covers basic concepts of cube and cuboid, painted cube/cuboid problems, formulas for finding cubes with 0, 1, 2, or 3 painted faces, volume and surface area calculations, and shortcut techniques for quick problem-solving.
About This Cheat Sheet
Complete guide to Cube and Cuboid for competitive exams:
Cube Basics
- Faces, Edges, Vertices
- Volume of Cube
- Surface Area of Cube
- Diagonal of Cube
Cuboid Basics
- Faces, Edges, Vertices
- Volume of Cuboid
- Surface Area of Cuboid
- Diagonal of Cuboid
Painted Cube Problems
- Painting Outer Surface
- Cutting into Smaller Cubes
- 0, 1, 2, 3 Painted Faces
- Formula-Based Solutions
Painted Cuboid Problems
- Painting Outer Surface
- Cutting into Smaller Cubes
- 0, 1, 2, 3 Painted Faces
- a×b×c Formula
Cube Cutting
- n×n×n Cube
- Number of Smaller Cubes
- Inner Cubes
- Face/Edge/Corner Cubes
Cuboid Cutting
- a×b×c Cuboid
- Number of Smaller Cubes
- Inner Cubes
- Face/Edge/Corner Cubes
Volume & Surface Area
- Volume Calculations
- Total Surface Area
- Lateral Surface Area
- Space Diagonal
Shortcut Techniques
- Quick Formulas
- Exam-Oriented Tricks
- Pattern Recognition
- Time-Saving Methods
Topics Covered
This cheat sheet covers all important Cube and Cuboid topics for quick revision:
| Topic | Description | Key Formula / Tip |
|---|---|---|
| Cube Properties | 6 faces (square), 12 edges (equal), 8 vertices | All sides equal (a = side) |
| Cuboid Properties | 6 faces (rectangle), 12 edges (not all equal), 8 vertices | Sides: a, b, c (length, breadth, height) |
| Volume of Cube | Space occupied by a cube | V = a³ |
| Surface Area of Cube | Total outer surface area | SA = 6a² |
| Volume of Cuboid | Space occupied by a cuboid | V = a × b × c |
| Surface Area of Cuboid | Total outer surface area | SA = 2(ab + bc + ca) |
| Diagonal of Cube | Space diagonal connecting opposite vertices | d = a√3 |
| Diagonal of Cuboid | Space diagonal connecting opposite vertices | d = √(a² + b² + c²) |
| Painted Cube - 3 Faces | Cubes with exactly 3 painted faces (corners) | Always 8 |
| Painted Cube - 2 Faces | Cubes with exactly 2 painted faces (edges) | 12(n-2) |
| Painted Cube - 1 Face | Cubes with exactly 1 painted face (face centers) | 6(n-2)² |
| Painted Cube - 0 Faces | Cubes with no painted faces (inner core) | (n-2)³ |
| Painted Cuboid - 3 Faces | Cubes with exactly 3 painted faces (corners) | Always 8 |
| Painted Cuboid - 2 Faces | Cubes with exactly 2 painted faces (edges) | 4[(a-2)+(b-2)+(c-2)] |
| Painted Cuboid - 1 Face | Cubes with exactly 1 painted face (face centers) | 2[(a-2)(b-2)+(b-2)(c-2)+(c-2)(a-2)] |
| Painted Cuboid - 0 Faces | Cubes with no painted faces (inner core) | (a-2)(b-2)(c-2) |
Key Formulas - Quick Reference
Cube Formulas (n×n×n)
| Parameter | Formula |
| Total Cubes | n³ |
| 3 Painted Faces | 8 |
| 2 Painted Faces | 12(n-2) |
| 1 Painted Face | 6(n-2)² |
| 0 Painted Faces | (n-2)³ |
Cuboid Formulas (a×b×c)
| Parameter | Formula |
| Total Cubes | a×b×c |
| 3 Painted Faces | 8 |
| 2 Painted Faces | 4[(a-2)+(b-2)+(c-2)] |
| 1 Painted Face | 2[(a-2)(b-2)+(b-2)(c-2)+(c-2)(a-2)] |
| 0 Painted Faces | (a-2)(b-2)(c-2) |
Volume & Surface Area
| Property | Formula |
| Cube Volume | a³ |
| Cube Surface Area | 6a² |
| Cube Diagonal | a√3 |
| Cuboid Volume | a×b×c |
| Cuboid Surface Area | 2(ab+bc+ca) |
| Cuboid Diagonal | √(a²+b²+c²) |
Cube vs Cuboid
| Property | Cube | Cuboid |
| Faces | 6 (all square) | 6 (rectangles) |
| Edges | 12 (equal) | 12 (not all equal) |
| Vertices | 8 | 8 |
| Volume | a³ | a×b×c |
| Surface Area | 6a² | 2(ab+bc+ca) |
Why This Cheat Sheet is Important
Master Painted Cube Problems
Learn to quickly find cubes with 0, 1, 2, or 3 painted faces using formulas.
Apply Shortcut Formulas
Use time-saving formulas for cube and cuboid problems in exams.
Understand Core Concepts
Build a strong foundation in cube and cuboid properties and calculations.
High Exam Weightage
Cube and cuboid questions appear frequently in quantitative aptitude sections.
Competitive Exams Covered
- SSC CGL
- SSC CHSL
- SSC MTS
- Bank PO & Clerk
- IBPS Exams
- Railway Recruitment Exams
- UPSC CSAT
- Defence Exams
- Insurance Exams
- CAT
- State Government Exams
Quick Exam-Oriented Examples
Question 1:
A cube of side 4 cm is painted on all its faces and then cut into smaller cubes of side 1 cm. How many smaller cubes have exactly 3 painted faces?
Step 1: Cube side = 4 cm, smaller cube side = 1 cm.
Step 2: n = 4 (number of divisions along each edge).
Step 3: Cubes with exactly 3 painted faces are at the 8 corners.
Formula: 3 Painted Faces = 8 (always for a cube)
Answer: 8 cubes
Question 2:
A cube of side 5 cm is painted on all its faces and then cut into smaller cubes of side 1 cm. How many smaller cubes have exactly 2 painted faces?
Step 1: Cube side = 5 cm, smaller cube side = 1 cm.
Step 2: n = 5 (number of divisions along each edge).
Step 3: Cubes with exactly 2 painted faces are on the edges (excluding corners).
Formula: 2 Painted Faces = 12(n-2)
Step 4: = 12(5-2) = 12 × 3 = 36
Answer: 36 cubes
Question 3:
A cube of side 6 cm is painted on all its faces and then cut into smaller cubes of side 1 cm. How many smaller cubes have exactly 1 painted face?
Step 1: Cube side = 6 cm, smaller cube side = 1 cm.
Step 2: n = 6 (number of divisions along each edge).
Step 3: Cubes with exactly 1 painted face are on the faces (excluding edges and corners).
Formula: 1 Painted Face = 6(n-2)²
Step 4: = 6(6-2)² = 6 × 4² = 6 × 16 = 96
Answer: 96 cubes
Question 4:
A cube of side 7 cm is painted on all its faces and then cut into smaller cubes of side 1 cm. How many smaller cubes have no painted faces?
Step 1: Cube side = 7 cm, smaller cube side = 1 cm.
Step 2: n = 7 (number of divisions along each edge).
Step 3: Cubes with no painted faces are the inner cubes (not on any face).
Formula: 0 Painted Faces = (n-2)³
Step 4: = (7-2)³ = 5³ = 125
Answer: 125 cubes
Question 5:
A cube of side 4 cm is painted on all its faces and then cut into smaller cubes of side 1 cm. Find the number of cubes with 3, 2, 1, and 0 painted faces.
Step 1: Cube side = 4 cm, smaller cube side = 1 cm.
Step 2: n = 4 (number of divisions along each edge).
Step 3: Apply formulas:
3 Faces: 8
2 Faces: 12(n-2) = 12(4-2) = 24
1 Face: 6(n-2)² = 6(4-2)² = 6×4 = 24
0 Faces: (n-2)³ = (4-2)³ = 2³ = 8
Total: 8 + 24 + 24 + 8 = 64 = 4³ ✓
Answer: 3 Faces: 8, 2 Faces: 24, 1 Face: 24, 0 Faces: 8
Question 6:
A cuboid of dimensions 5 cm × 4 cm × 3 cm is painted on all its faces and then cut into smaller cubes of side 1 cm. How many smaller cubes have no painted faces?
Step 1: Dimensions: a = 5, b = 4, c = 3 (all > 2).
Step 2: Cubes with no painted faces are the inner cubes.
Formula: 0 Painted Faces = (a-2)(b-2)(c-2)
Step 3: = (5-2)(4-2)(3-2) = 3 × 2 × 1 = 6
Answer: 6 cubes
Question 7:
A cuboid of dimensions 5 cm × 4 cm × 3 cm is painted on all its faces and then cut into smaller cubes of side 1 cm. How many smaller cubes have exactly 1 painted face?
Step 1: Dimensions: a = 5, b = 4, c = 3.
Formula: 1 Painted Face = 2[(a-2)(b-2)+(b-2)(c-2)+(c-2)(a-2)]
Step 2: (a-2)=3, (b-2)=2, (c-2)=1
Step 3: = 2[(3×2)+(2×1)+(1×3)]
Step 4: = 2[6+2+3] = 2×11 = 22
Answer: 22 cubes
Question 8:
A cuboid of dimensions 5 cm × 4 cm × 3 cm is painted on all its faces and then cut into smaller cubes of side 1 cm. How many smaller cubes have exactly 2 painted faces?
Step 1: Dimensions: a = 5, b = 4, c = 3.
Formula: 2 Painted Faces = 4[(a-2)+(b-2)+(c-2)]
Step 2: (a-2)=3, (b-2)=2, (c-2)=1
Step 3: = 4[3+2+1] = 4×6 = 24
Answer: 24 cubes
Question 9:
A cuboid of dimensions 5 cm × 4 cm × 3 cm is painted on all its faces and then cut into smaller cubes of side 1 cm. Find the number of cubes with 3, 2, 1, and 0 painted faces.
Step 1: Dimensions: a = 5, b = 4, c = 3.
Step 2: Apply formulas:
3 Faces: 8 (always)
2 Faces: 4[(a-2)+(b-2)+(c-2)] = 4[3+2+1] = 24
1 Face: 2[(a-2)(b-2)+(b-2)(c-2)+(c-2)(a-2)] = 2[6+2+3] = 22
0 Faces: (a-2)(b-2)(c-2) = 3×2×1 = 6
Total: 8 + 24 + 22 + 6 = 60 = 5×4×3 ✓
Answer: 3 Faces: 8, 2 Faces: 24, 1 Face: 22, 0 Faces: 6
Question 10:
Find the volume and total surface area of a cube with side length 5 cm.
Step 1: Side length a = 5 cm.
Step 2: Volume = a³ = 5³ = 125 cm³
Step 3: Total Surface Area = 6a² = 6×5² = 6×25 = 150 cm²
Answer: Volume = 125 cm³, Surface Area = 150 cm²
Question 11:
Find the space diagonal of a cuboid with dimensions 3 cm × 4 cm × 5 cm.
Step 1: Dimensions: a = 3, b = 4, c = 5.
Step 2: Space Diagonal = √(a² + b² + c²)
Step 3: = √(3² + 4² + 5²) = √(9 + 16 + 25) = √50 = 5√2 cm
Answer: 5√2 cm
Final Takeaway
Cube and Cuboid problems test your understanding of 3D geometry, visualization, and formula application. The key is to memorize the formulas for painted cube/cuboid problems, volume, surface area, and diagonal calculations.
Regular practice of cube and cuboid problems with different dimensions will significantly improve your speed and accuracy. This cheat sheet serves as a complete revision resource for mastering Cube and Cuboid questions in competitive examinations.
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Everything You Need for Cube and Cuboid Revision
Explore cube basics, cuboid basics, painted cube problems, painted cuboid problems, volume and surface area calculations, diagonal formulas, and shortcut techniques in one place for quick and effective revision.
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