Overview

The chapter “Rectangular Parallelepiped or Cuboid” (আয়তঘন) is an essential part of the Geometry section in the Class 10 Mathematics syllabus of the West Bengal Board. This chapter introduces students to three-dimensional (3D) geometry, helping them understand the structure, surface area, and volume of a solid figure called a cuboid. A clear conceptual understanding of a cuboid and its properties lays the foundation for advanced topics in mensuration and spatial geometry in higher classes.

A rectangular parallelepiped, commonly known as a cuboid, is a three-dimensional solid bounded by six rectangular faces. Each pair of opposite faces is equal and parallel. This chapter focuses on important formulae, properties, and applications related to cuboids in both real-world and academic contexts.

What is a Rectangular Parallelepiped or Cuboid?

A cuboid is a 3D geometric figure where all faces are rectangles. It has:

• 6 faces 
• 12 edges 
• 8 vertices 

In a rectangular cuboid, the angles between any two adjacent faces are right angles. Each face intersects the adjacent faces at 90 degrees. The shape closely resembles common objects like bricks, matchboxes, books, and boxes.

If the length, breadth, and height of a cuboid are l, b, and h respectively, then the geometry of the cuboid can be defined using these three dimensions.

Key Properties of a Cuboid

1. Faces, Edges, and Vertices

• A cuboid has 6 rectangular faces. 
• There are 12 edges, where each edge is a line segment connecting two adjacent vertices. 
• The 8 vertices are points where three edges meet. 

All faces are rectangular and opposite faces are equal in dimension. Every corner of a cuboid forms a right angle (90°).

2. Dimensions

The three primary dimensions of a cuboid are:

• Length (l) – The longest side 
• Breadth (b) – The shorter adjacent side 
• Height (h) – The vertical dimension 

These dimensions are used to calculate its surface area, lateral surface area, and volume.

Important Formulae in the Chapter

1. Total Surface Area (TSA) of a Cuboid

The total surface area refers to the sum of the areas of all six faces of the cuboid.

Formula:
TSA=2(lb+bh+hl)
Where,
l = length,
b = breadth,
h = height.

Example:
If a cuboid has length = 5 cm, breadth = 4 cm, and height = 3 cm,
then TSA = 2(5×4 + 4×3 + 3×5) = 2(20 + 12 + 15) = 2×47 = 94 cm²

2. Lateral Surface Area (LSA) of a Cuboid

The lateral surface area is the total area of the four vertical faces (excluding the top and bottom).

Formula:
LSA=2h(l+b)

Example:
For a cuboid with length = 6 cm, breadth = 4 cm, height = 5 cm,
LSA = 2 × 5 × (6 + 4) = 10 × 10 = 100 cm²

3. Volume of a Cuboid

The volume represents the total space occupied inside the cuboid.

Formula:
Volume=l×b×h

Example:
Length = 8 cm, Breadth = 5 cm, Height = 2 cm,
Volume = 8 × 5 × 2 = 80 cm³

Diagonal of a Cuboid

The space diagonal of a cuboid is the longest diagonal stretching from one corner to the opposite corner through the interior of the cuboid.

Formula:
Diagonal=l2+b2+h2

Example:
If l = 3 cm, b = 4 cm, h = 12 cm,
Diagonal = √(9 + 16 + 144) = √169 = 13 cm

Units of Measurement

Always ensure that all dimensions are in the same unit before applying any formula. The standard units used are:

• Length, breadth, height: cm, m, mm 
• Area: cm², m² 
• Volume: cm³, m³ 

Conversion Tip:
1 m = 100 cm
1 m² = 10,000 cm²
1 m³ = 1,000,000 cm³

Real-Life Applications of Cuboids

Understanding the concept of cuboids is highly useful in daily life. Common applications include:

• Calculating the packaging size for boxes and containers. 
• Measuring the amount of material required to construct a rectangular room. 
• Determining the storage capacity of tanks or cabinets. 
• Estimating shipping volume and surface area for covering objects. 

Example:
A wooden box is 2 m long, 1.5 m wide, and 1 m high. Its volume = 2 × 1.5 × 1 = 3 m³. This tells us the box can hold materials with a total volume of 3 cubic metres.

Summary of Key Concepts

Property	Formula
Total Surface Area (TSA)	2(lb + bh + hl)
Lateral Surface Area	2h(l + b)
Volume	l × b × h
Diagonal	√(l² + b² + h²)

Conclusion

The chapter “Rectangular Parallelepiped or Cuboid” (আয়তঘন) helps Class 10 students of the West Bengal Board develop a clear understanding of three-dimensional figures. By mastering surface area, volume, and diagonal calculations, students can solve both theoretical and practical problems with confidence.

It is crucial to practice all formula-based problems, understand unit conversions, and apply logical reasoning while solving geometry questions. This chapter also forms the base for complex mensuration problems in future academic studies.

For better results, students should focus on using the correct formula, maintain dimensional accuracy, and apply the concepts to practical scenarios effectively.