TEACHGYAN • CLASS 8 MATHEMATICS • GANITA PRAKASH PART II
Chapter 14: Area
NCERT-aligned detailed notes • Concepts • Worked examples • MCQs • Competency • HOTS
📚 NCERT Topic-by-Topic Explanation
1. Area of rectangles and squares
Area measures the amount of two-dimensional surface inside a closed figure. A rectangle of length l and breadth b has area l×b; a square of side s has area s².
🧮 Worked Example
Start with the simplest case, apply the rule once, record the result, and repeat the same rule to identify the pattern.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
2. Area of triangles
A triangle with base b and perpendicular height h has area 1/2×b×h. The height must be perpendicular to the chosen base; a slanting side is not automatically the height.
🧮 Worked Example
Example: For data 5, 7, 8, 10, the mean is (5+7+8+10)÷4=30÷4=7.5. The total deviation from 7.5 is zero, showing the balancing interpretation.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
3. Area of parallelograms
A parallelogram has area b×h. If a triangular piece is shifted from one side to the other, the parallelogram can be rearranged into a rectangle with the same area.
🧮 Worked Example
Example: Let a number be x. If a trick says multiply by 3 and add 6, the result is 3x+6. Reversing the steps gives x=(result−6)÷3.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
4. Area of polygons by decomposition
A polygon can often be split into triangles, rectangles or parallelograms. Add the areas of non-overlapping parts, or subtract an unwanted part from a larger figure.
🧮 Worked Example
Example: A parallelogram has base 12 cm and perpendicular height 7 cm. Its area is 12×7=84 cm². The perpendicular height, not the sloping side, is used.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
5. Area of rhombus
A rhombus can be treated through its diagonals or by decomposing it into triangles. The formula A=1/2×d₁×d₂ follows from the geometry of those parts.
🧮 Worked Example
Example: A parallelogram has base 12 cm and perpendicular height 7 cm. Its area is 12×7=84 cm². The perpendicular height, not the sloping side, is used.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
6. Choosing a base and perpendicular height
Always check units: cm², m² and so on. If a calculated area is wildly inconsistent with the dimensions, recheck the formula, height and unit conversion.
🧮 Worked Example
Example: A parallelogram has base 12 cm and perpendicular height 7 cm. Its area is 12×7=84 cm². The perpendicular height, not the sloping side, is used.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
7. Composite figures and problem solving
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🧮 Worked Example
Example: A parallelogram has base 12 cm and perpendicular height 7 cm. Its area is 12×7=84 cm². The perpendicular height, not the sloping side, is used.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
8. Unit conversion and reasonableness
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🧮 Worked Example
Example: A parallelogram has base 12 cm and perpendicular height 7 cm. Its area is 12×7=84 cm². The perpendicular height, not the sloping side, is used.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
9. Deriving and applying area formulas
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🧮 Worked Example
Example: A parallelogram has base 12 cm and perpendicular height 7 cm. Its area is 12×7=84 cm². The perpendicular height, not the sloping side, is used.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
🧠 Concept-Based MCQs
- The area of a rectangle 8 cm by 5 cm is: A) 13 cm² B) 26 cm² C) 40 cm² D) 80 cm²
- Triangle area uses: A) base×height B) 1/2×base×perpendicular height C) side×side D) diagonal×diagonal
- The height of a parallelogram must be: A) any side B) perpendicular to the chosen base C) diagonal always D) longest side
Answers: 1-B, 2-B, 3-A.
📌 Competency-Based Case Study
A learner is given an unfamiliar diagram, data display or puzzle and remembers formulas but is unsure which idea to use. The first step is to identify the structure and explain the relationship before calculating.
- What information is given?
- Which NCERT concept applies?
- Show the reasoning step by step.
- How can the answer be checked?
🔥 HOTS & Competition Corner
- Find a pattern and state a rule that predicts the next stage.
- Create a counterexample to a tempting but false claim.
- Solve the problem in two different ways and compare the methods.
- Design your own NCERT-style challenge question.
✍️ Exam-Ready Practice
- Explain the central concept in your own words.
- Solve a direct application problem.
- Solve an unfamiliar competency-based problem.
- Explain one common mistake and correct it.
- Write a complete 5-mark solution with working and conclusion.
🏆 Five-Level Mastery
| Level | Skill |
|---|---|
| 1 Recall | Definitions, terms and formulas |
| 2 Understand | Explain concepts and interpret examples |
| 3 Apply | Solve routine and real-life problems |
| 4 Analyse | Detect errors, compare methods and justify |
| 5 Create | Build a pattern, puzzle, model or problem |
🔁 One-Minute Revision
Concept → Condition → Representation → Method → Example → Check → Application.
✅ Mastery Checklist
- ☐ I can explain every topic.
- ☐ I can solve standard problems.
- ☐ I can apply the concept in unfamiliar situations.
- ☐ I can justify my method.
- ☐ I can find and correct mistakes.