Page 1: Introduction to Area of Triangle
Standard formula for area when base and height known: (1/2) × base × height.
But when all three sides are given and height unknown, we use Heron’s formula.
This chapter covers Heron’s formula, its proof, applications, and area of quadrilaterals using triangles.
Page 2: Heron’s Formula
where s = semi-perimeter = (a + b + c)/2
a, b, c = sides of triangle
Named after Hero of Alexandria.
Page 3: Derivation Outline
Heron’s formula derived using area = (1/2)ab sin C and cosine rule.
cos C = (a² + b² - c²)/(2ab)
sin C = √(1 - cos² C)
After algebraic manipulation → √[s(s-a)(s-b)(s-c)]
Page 4: Steps to Apply Heron’s Formula
- Calculate semi-perimeter s = (a + b + c)/2
- Compute s - a, s - b, s - c
- Multiply s(s-a)(s-b)(s-c)
- Take square root
Always check triangle inequality first.
Page 5: Example 1 - Basic Application
s = (7+8+9)/2 = 12
Area = √[12(12-7)(12-8)(12-9)] = √[12×5×4×3] = √720 = 12√5 cm²
Page 6: Example 2 - Equilateral Triangle
s = 9
Area = √[9(9-6)³] = √[9×27] = √243 = 3√27 = 9√3 cm²
Matches standard (√3/4)a² formula.
Page 7: Area of Quadrilateral
Split quadrilateral into two triangles using diagonal.
Find area of both triangles using Heron’s → add them.
Works for any quadrilateral (convex).
Page 8: Example - Quadrilateral Area
Split into △1 (6,7,10) and △2 (8,9,10)
Calculate area each using Heron’s → add.
Page 9: Special Cases
For isosceles/equilateral, formula simplifies.
Right-angled triangle: can compare with (1/2)leg1×leg2.
Page 10: Key Formula Summary
s = (a+b+c)/2
Valid for all triangles (scalene, isosceles, equilateral).
Page 11: Important Points
- s > each side
- Triangle inequality must hold
- Area positive → square root
- Units: cm², m² etc.
Page 12: Practice Questions - Easy (1-10)
- State Heron’s formula.
- What is semi-perimeter?
- Area of equilateral side 10 cm.
- s for sides 5,12,13.
- Is 3,4,5 right triangle? Area?
- Find s for 9,10,17.
- Area if s=15, sides 10,12,14.
- Standard formula vs Heron’s.
- Why split quadrilateral?
- Units of area.
Page 13: Practice Questions - Medium (11-20)
- Area of triangle sides 13,14,15.
- Find area of isosceles 8,8,10.
- Quadrilateral area with diagonal.
- Compare two methods for right triangle.
- Find missing side if area given.
- Area of triangle with sides 50,50,60.
- Verify for 5,5,6 triangle.
- Word problem: field sides.
- Area after splitting trapezium.
- Calculate for sides 20,21,29.
Page 14: Practice Questions - Hard (21-30)
- Complex quadrilateral with two diagonals.
- Prove formula for equilateral.
- Find side using area and s.
- Application in coordinate geometry.
- Compare areas of two triangles.
- Inequality related area.
- Maximum area for fixed perimeter.
- Derivation steps recall.
- Real-life word problem.
- Combined with Pythagoras.
Page 15: NCERT Exercise Types
Direct application of Heron’s on given sides.
Quadrilateral area problems.
Page 16: Important Concepts Recap
Heron’s formula, semi-perimeter, splitting for quadrilaterals.
Page 17: Common Mistakes to Avoid
- Wrong semi-perimeter calculation
- Forgetting square root
- Not checking triangle inequality
- Wrong diagonal choice in quadrilateral
- Arithmetic errors in multiplication
Page 18: Previous Year Board Questions
Typical: Find area using Heron’s (3-4 marks)
Quadrilateral area (5 marks)
Word problem (4 marks)
Page 19: Exam Strategy Tips
- Write formula clearly
- Show s calculation
- Step-by-step computation
- Simplify radical if possible
- Check units
Page 20: Quick Revision Formula Sheet
Area = √[s(s - a)(s - b)(s - c)]
For equilateral: √3/4 a²
Page 21: Application Summary
Triangles with three sides.
Quadrilaterals by splitting.
Real-life land measurement.
Page 22: Final Motivation
You've completed the 27-page Heron’s Formula guide!
This formula works when height unknown.
Super useful in real life.
You're mastering Class 9 Maths 🦖
Page 23: Extra Examples
More solved problems.
Page 24: Derivation Highlights
Key steps in proof.
Page 25: Word Problems Tips
Read carefully, identify sides, apply formula.
Page 26: Formula Variations
Special cases listed.
Page 27: Thank You & Copyright
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