Category: Verbal Reasoning

Topic: Cube and Cuboid

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Cube and Cuboid

🧠 Build a strong foundation in logical reasoning with clear explanations and real-world examples. Understand core concepts and develop critical thinking skills.

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Introduction & Key Concepts

Study Material

Cube and Cuboid

Cube and Cuboid is one of the most important topics in Logical Reasoning and Non-Verbal Reasoning sections of competitive examinations. These questions test visualization ability, spatial reasoning, analytical thinking, observation skills, and logical interpretation.

Questions from Cube and Cuboid are commonly asked in Banking, SSC, Railway, Insurance, Defence, MBA entrance, and various government examinations.

What is a Cube?

A Cube is a three-dimensional solid structure in which all edges are equal in length.

A Cube has:

6 Faces
12 Edges
8 Vertices (Corners)

All faces of a cube are squares.

Visual Structure of a Cube

        +--------+
       /        /|
      /        / |
     +--------+  |
     |        |  |
     |        |  +
     |        | /
     |        |/
     +--------+

Important Properties of a Cube

Property	Value
Faces	6
Edges	12
Vertices	8
Shape of Faces	Square
Length = Breadth = Height	Equal

What is a Cuboid?

A Cuboid is a three-dimensional solid figure having rectangular faces.

A Cuboid has:

6 Faces
12 Edges
8 Vertices

Opposite faces of a cuboid are equal rectangles.

Visual Structure of a Cuboid

        +------------+
       /            /|
      /            / |
     +------------+  |
     |            |  |
     |            |  +
     |            | /
     |            |/
     +------------+

Difference Between Cube and Cuboid

Cube	Cuboid
All sides equal	Length, breadth, height may differ
Faces are squares	Faces are rectangles
Perfect symmetry	Rectangular symmetry
All edges same length	Edges may differ

Concept of Smaller Cubes

Large cubes are often divided into smaller equal cubes in reasoning questions.

If a cube is divided equally into:

n × n × n

Then:

Total Smaller Cubes = n³

Example of Smaller Cubes

If each edge contains:

5 smaller cubes

Then total cubes formed:

5³ = 125

Minimum Number of Cuts

Questions frequently ask the minimum cuts required to divide a cube into smaller cubes.

For:

n × n × n cubes

Minimum cuts required:

3(n − 1)

Example – Minimum Cuts

If a cube is divided into:

5 × 5 × 5 cubes

Minimum cuts:

3(5 − 1) = 12

Painted Cube Concept

One of the most important concepts in Cube and Cuboid reasoning is painted cubes.

A large cube is painted from outside and then divided into smaller cubes.

Questions are asked about:

Cubes with 3 faces painted
Cubes with 2 faces painted
Cubes with 1 face painted
Cubes with no face painted
Visible cubes

Cube with Three Faces Painted

Corner cubes always have three faces painted.

Important Rule:

Always 8 Cubes

Every cube has 8 corners.

Cube with Two Faces Painted

Edge cubes (excluding corners) have two faces painted.

Formula:

12(n − 2)

Cube with One Face Painted

Face-centre cubes have only one painted face.

Formula:

6(n − 2)²

Cube with No Face Painted

Interior cubes remain completely unpainted.

Formula:

(n − 2)³

Visible Cubes from Outside

Visible cubes are cubes seen from outside after painting.

Formula:

n³ − (n − 2)³

Painted Cube Formula Summary

Type of Cube	Formula
3 Faces Painted	8
2 Faces Painted	12(n − 2)
1 Face Painted	6(n − 2)²
No Face Painted	(n − 2)³
Visible Cubes	n³ − (n − 2)³

Concept of Cube Folding

Cube Folding questions involve folding a flat figure into a cube.

Candidates must identify:

Opposite faces
Adjacent faces
Matching cube structures
Incorrect cube formations

Important Concepts in Cube Folding

Opposite Faces

Opposite faces never touch each other.

Adjacent Faces

Adjacent faces always share an edge.

Cube Rotation

Cube orientation changes after rotation.

Net Structure

Flat patterns form 3D cubes after folding.

Most Important Skills Required

Visualization Skill
Spatial Reasoning
Analytical Thinking
Observation Ability
Pattern Recognition
Logical Interpretation

Common Types of Cube and Cuboid Questions

Question Type	Description
Painted Cubes	Face-color counting questions
Cube Cutting	Division into smaller cubes
Minimum Cuts	Least cuts required
Cube Folding	Net-to-cube formation
Visible Cubes	Outer cube counting
Opposite Faces	Finding opposite sides

Common Mistakes in Cube and Cuboid Questions

Ignoring corner cubes
Confusing edge cubes with face cubes
Wrong formula application
Incorrect cube rotation visualization
Ignoring interior cubes
Confusing adjacent and opposite faces

Quick Solving Framework

Identify Cube Type
        │
        ▼
Determine Dimensions
        │
        ▼
Apply Correct Formula
        │
        ▼
Analyze Faces & Edges
        │
        ▼
Verify Final Result

Final Takeaway

Cube and Cuboid is a highly important reasoning topic based on spatial visualization, logical interpretation, cube division, painted-face analysis, and three-dimensional reasoning concepts.

Strong understanding of painted cubes, cube cutting, minimum cuts, visible cubes, and cube folding significantly improves solving speed and logical accuracy in competitive examinations.

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