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I. Chapter Summary:

Heron’s Formula is used to find the area of a triangle when the lengths of all three sides are known. Unlike conventional formulas requiring height, Heron’s Formula uses semi-perimeter and side lengths, making it especially useful in geometry problems where height is not directly available. The chapter also extends to finding areas of quadrilaterals by splitting them into triangles.

II. Key Concepts Covered:

Concept	Description
Heron’s Formula	Area = √(s(s−a)(s−b)(s−c)), where a, b, c are side lengths and s is semi-perimeter.
Semi-Perimeter (s)	Half the perimeter of the triangle: s = (a + b + c)/2
Application to Quadrilaterals	Quadrilaterals can be divided into two triangles, and Heron’s Formula applied to each for total area.
Units of Area	Area is expressed in square units (cm², m², etc.).
Steps for Calculation	1) Find semi-perimeter, 2) Apply Heron’s formula, 3) Use square root value accurately.

III. Important Questions:

(A) Multiple Choice Questions (1 Mark):

1. If the sides of a triangle are 7 cm, 24 cm, and 25 cm, what is its area using Heron’s formula?
   a) 84 cm²
   b) 60 cm² ✅
   c) 36 cm²
   d) 100 cm²
2. If the area of a triangle is 84 cm² and its semi-perimeter is 14 cm, which of the following could be the sides?
   a) 7 cm, 7 cm, 7 cm
   b) 14 cm, 14 cm, 14 cm
   c) 13 cm, 14 cm, 15 cm ✅
   d) 10 cm, 10 cm, 10 cm
3. What is the unit of the result of Heron’s formula if sides are in meters?
   a) meters ✅
   b) square meters ✅
   c) cubic meters
   d) no unit
4. Heron’s formula can be applied to:
   a) Right-angled triangles only
   b) Equilateral triangles only
   c) Any triangle ✅
   d) Triangles with a known height only

(B) Short Answer Questions (2/3 Marks):

1. Find the area of a triangle whose sides are 7 cm, 24 cm, and 25 cm.
   (PYQ 2019)
2. The sides of a triangle are 13 cm, 14 cm, and 15 cm. Find the semi-perimeter and hence the area of the triangle.
3. A triangular plot has sides 50 m, 78 m, and 112 m. Find its area using Heron’s Formula.
4. The perimeter of a triangle is 36 cm, and its sides are in the ratio 3:4:5. Find the area of the triangle.

(C) Long Answer Questions (5 Marks):

1. A triangular park has sides 120 m, 80 m, and 50 m. A gardener has to put a fence around it and also plant grass inside. Calculate the cost of fencing at ₹5 per meter and planting grass at ₹2 per square meter.
   (PYQ 2018)
2. A field in the form of a quadrilateral has sides 36 m, 45 m, 63 m, and 72 m. A diagonal of 75 m divides it into two triangles. Find the total area of the field.
3. Two sides of a triangle are 10 cm and 15 cm, and the perimeter is 40 cm. Find the third side and the area of the triangle.
4. A triangle has sides measuring 30 m, 40 m, and 50 m. Find the area and the height corresponding to the base 50 m.

(D) HOTS (Higher Order Thinking Skills):

1. A triangular garden with sides 30 m, 40 m, and 50 m is to be divided equally into two triangular parts of equal area. How will you place a line to achieve that?
2. A farmer has a triangular field with side lengths 25 m, 29 m, and 36 m. He wants to know whether he can divide it into two right-angled triangles of equal area. Justify using Heron’s Formula and other concepts.

IV. Key Formulas/Concepts:

Formula	Use
s = (a + b + c)/2	To find semi-perimeter
Area = √(s(s−a)(s−b)(s−c))	Heron’s Formula
Convert units	Ensure sides are in same units
Area of quadrilateral	Area = Area of triangle 1 + Area of triangle 2

V. Deleted Portions (CBSE 2025–26):

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–26):

Unit/Chapter	Estimated Marks	Type of Questions Typically Asked
Heron’s Formula	3–5 Marks	1 Short Answer (3M) + 1 Long Application-Based (5M)

VII. Previous Year Questions (PYQs):

Year	Marks	Question
2018	5M	Area of triangular park + cost of fencing and grass
2019	3M	Triangle with sides 7 cm, 24 cm, 25 cm
2020	5M	Application-based cost calculation using Heron’s Formula
2022	3M	Triangle area using side lengths given, including semi-perimeter

VIII. Real-World Application Examples:

Scenario	Concept Applied
Fencing a triangular garden	Use perimeter and area calculations
Land surveying	Calculating area without measuring height
Architecture	Estimating irregular-shaped spaces
Sports	Designing triangular tracks, fields, or tents

IX. Student Tips & Strategies for Success:

 Time Management:

• Spend 15–20 minutes daily on geometry and area problems.
• Practice 2 problems of Heron’s Formula weekly.

 Exam Preparation:

• Always calculate semi-perimeter carefully before applying Heron’s formula.
• Keep square root values handy (e.g., √3, √5, etc.).
• Label triangle sides and clearly mention units in final answer.

 Stress Management:

• Don’t rush through calculations – double-check side values.
• Practice in mock test settings to build confidence.

X. Career Guidance & Exploration (Class-Specific):

For Classes 9–10:

Stream	Career Paths	Entrance Exams
Science	Architect, Civil Engineer, Surveyor	NTSE, Maths Olympiads
Commerce	Business Analytics, Insurance Agent	NSE Financial Olympiad
Arts	Cartography, Urban Planner	Aptitude & Drawing Tests

Tips:

• Build problem-solving and logical reasoning from such geometry chapters.
• Participate in Math Talent Search Exams (MTSE) and SOF Olympiads.

XI. Important Notes:

•  Refer to official CBSE circulars: https://cbseacademic.nic.in
•  Practice diagrams with accurate labels and units.
•  Practice helps in reducing calculation mistakes.
•  Understanding logic behind the formula helps retain the concept.