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📘 Class 9 Maths Chapter 10 – Heron’s Formula | Full Chapter Summary

In this chapter, "Heron’s Formula", we learn how to calculate the area of a triangle when all three sides are known, without the need for height or altitude. This formula, discovered by Hero of Alexandria, is very useful in solving real-life and exam problems involving triangles and quadrilaterals.


🔹 1. Area of a Triangle – Usual Methods
Before Heron’s Formula, the area of a triangle was calculated using:


½ × base × height (when height is known).


½ × ab × sin C (when two sides and included angle are known).


But when only three sides are given, we use Heron’s Formula.


🔹 2. Heron’s Formula
For a triangle with sides a, b, c:

Semi-perimeter, s= (a+b+c)/2

Area,


A = √{s(s-a)(s-b)(s-c)}



🔹 3. Steps to Use Heron’s Formula

Find the semi-perimeter .

Substitute the values of .

Apply under the square root and simplify.

Final result gives the area in square units.


🔹 4. Applications of Heron’s Formula

✅ Area of a Triangle (when three sides are known).
 

✅ Area of Isosceles/Scalene Triangles (without height).
 

✅ Area of Quadrilaterals – By dividing them into two triangles and applying Heron’s Formula separately.
 

✅ Real-life Problems – Finding plots of land, fields, and triangular designs.


🔹 5. Extension – Area of Quadrilaterals

If a quadrilateral can be divided into two triangles (by drawing a diagonal), then:

Calculate the area of each triangle using Heron’s Formula.

Add them to get the total area of the quadrilateral.


🎯 Key Learning Outcomes

Understanding the derivation and application of Heron’s Formula.

Finding the area of triangles without knowing the height.

Solving real-life application problems.

Extending the formula to find the area of quadrilaterals.

Building problem-solving skills for geometry-based questions.


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