Introduction to Surface Areas
Introduction of surface areas of cube and cuboid.
What is Chapter 14 — Surface Area and Volume About?
Chapter 14 — Surface Areas and Volume of Class 8 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) introduces one of the most practically useful topics in geometry: calculating how much material covers the outside of a solid (surface area) and how much space it occupies inside (volume). In this introduction, the focus is specifically on two fundamental 3-D shapes — the Cuboid and the Cube.
You encounter these shapes everywhere in daily life — a cardboard box is a cuboid, a dice is a cube, a room is a cuboid, an ice-cube is (approximately) a cube. Understanding how to measure their surfaces and interiors is essential for real-world applications like packaging, construction, and manufacturing, and it is a high-scoring topic in board exams.
What is Surface Area? — The Net Approach
The surface area of a solid is the total area of all its outer faces taken together. The smartest way to understand this is through a net — the flat shape you get when you unfold the solid completely and lay it flat on paper.
Imagine cutting along some edges of a cardboard box and unfolding it. The flat shape you get is the net. If you measure the area of every rectangle in that net and sum them up, you get the total surface area of the box.
Length l, Breadth b, Height h
Has 6 rectangular faces in 3 pairs
All 6 faces may have different dimensions
All sides equal — side length l
Has 6 identical square faces
Every face has area = l²
The Six Faces of a Cuboid — Step by Step
A cuboid has 6 rectangular faces arranged in 3 opposite pairs. When you unfold a cuboid into its net, you see all 6 faces laid flat. The faces are labelled I through VI, and each pair shares the same dimensions:
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Face V l × b |
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|
Face I l × h |
Face II b × h |
Face III l × h |
Face IV b × h |
|
Face VI l × b |
Each face and its area:
Total Surface Area of a Cuboid — Derivation
The total surface area (TSA) is the sum of the areas of all six faces. Adding all face areas from the net:
Lateral Surface Area of a Cuboid — Derivation
The lateral surface area (LSA) — also called the curved surface area or side surface area — counts only the four side faces (I, II, III, IV) and excludes the top and bottom. This is useful when you need to paint only the walls of a room (ignoring floor and ceiling), or wrap the sides of a box.
Total Surface Area of a Cube — Derivation
A cube has all sides equal to l, so every face is a square with area l². The net of a cube shows 6 identical square panels.
| Top l² |
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| Left l² |
Front l² |
Right l² |
| Bottom l² |
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| Back l² |
Lateral Surface Area of a Cube — Derivation
The lateral surface area of a cube counts only the four vertical side faces, leaving out the top and bottom squares. Since all four side faces of a cube are identical squares with area l², the derivation is straightforward:
All Four Key Formulas — Quick Reference
Memorising all four formulas together is the most efficient exam strategy. The table below shows all formulas for both cuboid and cube, side by side:
| Type of Area | Cuboid (l × b × h) | Cube (side = l) |
|---|---|---|
| Total Surface Area (TSA) | 2(lh + bh + lb) | 6l² |
| Lateral Surface Area (LSA) | 2h(l + b) | 4l² |
| Area of one face | Depends on which face | l² |
| Number of faces | 6 | 6 |
| Unit of surface area | Square units — cm², m², etc. | |
TSA Cuboid = 2(lh + bh + lb) | LSA Cuboid = 2h(l + b)
TSA Cube = 6l² | LSA Cube = 4l²
Worked Examples — Applying the Formulas
This tells you how much paint is needed to cover the four walls of the room (not the ceiling or floor).
Real-World Applications of Surface Area
- Painting walls of a room — Use LSA of cuboid (4 walls only). Add the ceiling area (l × b) if needed, but not the floor.
- Wrapping a gift box — Use TSA of cuboid to find how much wrapping paper is needed. Always add a little extra for overlaps.
- Tin/sheet metal work — When making a metal box (like a tiffin box or storage container), the total sheet required = TSA of the cuboid.
- Tiling a swimming pool — The walls and floor of a rectangular pool need tiles; this uses LSA + area of the base.
- Packaging design — Industries calculate TSA to minimise the material used in making boxes, reducing cost and waste.
Common Mistakes to Avoid
- Using wrong formula: Confusing TSA = 2(lh + bh + lb) with LSA = 2h(l + b). Remember — TSA includes the top and bottom faces; LSA does not.
- Wrong substitution order: In a cuboid, l, b, h can be assigned to any dimension as long as they are consistent. Swapping l and b is fine — but make sure you use the same values throughout.
- Forgetting to square in cube formulas: For a cube of side 6, TSA = 6 × 6² = 6 × 36 = 216, NOT 6 × 6 = 36.
- Forgetting the factor of 2: Students sometimes write TSA = lh + bh + lb (missing the ×2 factor). The full formula is 2(lh + bh + lb).
- Unit errors: Area is always in square units. If l is in cm, TSA is in cm² — never just cm.
Formula Derivation Summary — How It All Connects
Every surface area formula in this chapter comes from the same net-based reasoning. Here is a compact summary of how each formula is derived, which is useful for long-answer questions in Telangana and Andhra Pradesh board exams:
| Shape | Type | Faces Counted | Derivation | Final Formula |
|---|---|---|---|---|
| Cuboid | TSA | All 6 faces | lh + bh + lh + bh + lb + lb = 2lh + 2bh + 2lb | 2(lh + bh + lb) |
| Cuboid | LSA | 4 side faces only | lh + bh + lh + bh = 2lh + 2bh = 2h(l+b) | 2h(l + b) |
| Cube | TSA | All 6 faces | l² × 6 (since all faces are equal) | 6l² |
| Cube | LSA | 4 side faces only | l² × 4 (four side squares) | 4l² |
What This Introduction Prepares You For
The surface area concepts introduced here form the base for all subsequent exercises in Chapter 14. In the exercises, you will apply the formulas TSA = 2(lh + bh + lb) and LSA = 2h(l + b) to a wide range of word problems involving painting, tiling, wrapping, and manufacturing. You'll also learn the formula for Volume of Cube and Cuboid — how much space the solid occupies — which is the natural next concept after surface area.
These ideas extend directly into Class 9 and Class 10, where you will compute surface areas of cylinders, cones, and spheres using the same net-based approach. For CBSE, Telangana, and Andhra Pradesh board exams, surface area questions are high-frequency, often carrying 4–5 marks and appearing in both short-answer and long-answer sections.
A strong understanding of Chapter 13 — Visualizing 3-D in 2-D (faces, edges, vertices, and nets) is the perfect foundation for this chapter. If you are comfortable identifying the net of a cuboid, computing its surface area becomes completely natural.