Derivation of Surface Area of a Cuboid

A cuboid is a three-dimensional shape with six rectangular faces, each having length (l), breadth (b), and height (h).

Step 1: Understanding the Faces of a Cuboid

A cuboid has 6 faces, grouped as follows:

1. Two identical length × breadth faces → Area = l×b
2. Two identical breadth × height faces → Area = b×h
3. Two identical length × height faces → Area = l×h

Step 2: Summing Up the Areas of All Faces

Total Surface Area=2(l×b)+2(b×h)+2(l×h)

Step 3: Factoring Out the Common Term

Total Surface Area=2(lb+bh+hl)

Thus, the formula for the total surface area of a cuboid is:

Surface Area=2(lb+bh+hl)​

Key Takeaways:

• The surface area represents the total area covered by all six faces.
• It is used to calculate the amount of material needed to cover a cuboid-shaped object.

• Derivation of Surface Area of a Cube

•  

  A cube is a special type of cuboid where all sides are equal. If each edge of the cube is l ,then:

  Step 1: Understanding the Faces of a Cube

  • A cube has 6 identical square faces.
  • Each face is a square with side length = l
  • The area of one square face = l×l=l²

  Step 2: Summing Up the Areas of All Faces

  Total Surface Area=6×l²

  Thus, the formula for the total surface area of a cube is:

  Surface Area=6l²

  • The lateral surface area refers to only the four vertical faces, excluding the top and bottom.
  • Since each face has an area of l² and there are 4 lateral faces, we get:
    

     

• Derivation of Surface Area of a Cylinder

  A cylinder is a three-dimensional shape with two circular bases and a curved lateral surface. The total surface area of a cylinder includes:

  

•  

  1. The curved (lateral) surface area (CSA)
  2. The area of the two circular bases

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  1. Curved Surface Area (CSA) or Lateral Surface Area (LSA)

  • Imagine unrolling the curved surface of a cylinder—it forms a rectangular strip.

  • The height of the rectangle is equal to the height (h) of the cylinder.

  • The width of the rectangle is equal to the circumference of the base (which is a circle with radius r):

    Circumference of base=2πr

  • Thus, the curved surface area (CSA) is:

    CSA=Circumference×Height=2πrh

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  2. Area of the Two Circular Bases

  • Each base is a circle with radius r

  • The area of one circular base is:

    πr²

  • Since the cylinder has two bases, the total area of both bases is:

    2πr²

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  3. Total Surface Area (TSA) of a Cylinder

  The total surface area (TSA) is the sum of:

  1. Curved Surface Area (CSA)
  2. Two Circular Base Areas

  TSA= CSA+Area of both base

• A=2πrh+2πr²
• =2Πr(h+r)

• Final Formulas

  ✅ Curved Surface Area (CSA):

  CSA=2πrh

  ✅ Total Surface Area (TSA):

  TSA=2πr(r+h)

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  Key Points:

  • CSA (or LSA) = Only the curved side area.
  • TSA = Curved Surface + Two Base Areas.
  • Used in bottle designs, pipes, tanks, and packaging calculations.

• What is Lateral Surface?

  The lateral surface of a 3D solid shape refers to the area of all its side faces, excluding the top and bottom faces. It is also known as the curved surface area (CSA) in some cases, like for cylinders and cones.

  Lateral Surface Area (LSA) of Different Shapes

  1. Cube

     • A cube has 4 vertical faces (excluding the top and bottom).
     • Each face has an area of a2a^2a2, where a is the side length.
     • Formula: LSA = 4a²

  2. Cuboid

     • A cuboid has 4 vertical rectangular faces (excluding top and bottom).
     • Formula: LSA=2h(l+b)
     • where l= length, b= breadth, h= height)

  3. Cylinder

     • The lateral surface of a cylinder is its curved surface (side surface).
     • Formula: LSA=2πrh (where r = radius, h= height)

• Summary of Surface Area Formulas

  Shape	Lateral/Curved Surface Area (LSA/CSA)	Total Surface Area (TSA)
  Cube	4a²	6a²
  Cuboid	2h(l+b)	2(lb+bh+hl)
  Cylinder	2πrh	2πr(r+h)

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  Explanation of Symbols

  • a = Side length of cube
  • l = Length, b = Breadth, h= Height of cuboid
  • r = Radius of cylinder base, h = Height of cylinder

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  Key Differences

  • Lateral/Curved Surface Area (LSA/CSA): Only the side (excluding top & bottom).
  • Total Surface Area (TSA): Includes both LSA and the base(s).