CBSE Class 9 Maths Notes: Herons Formula

**Heron’s Formula for the Area of a Triangle**

Heron’s Formula provides a method to calculate the area of a triangle when the lengths of all three sides are known. It’s especially useful when the height of the triangle isn’t readily available.

This formula bypasses the need for knowing the height or any of the interior angles of a triangle.

Definitions

Before we delve into the formula, let’s define the key components:

• Sides: Let the sides of the triangle be denoted as *a*, *b*, and *c*. These represent the lengths of the three sides.
• Semi-Perimeter (s): The semi-perimeter is half of the triangle’s perimeter. It is calculated as: $s = \frac{a + b + c}{2}$.

Core Principles

The foundation of Heron’s formula lies in relating the area of a triangle directly to its side lengths. It employs the semi-perimeter to simplify the area calculation.

Formulas

The core formula is:

Area of Triangle = $\sqrt{s(s-a)(s-b)(s-c)}$

Examples

Let’s consider a triangle with sides *a* = 13 cm, *b* = 14 cm, and *c* = 15 cm.

1. Calculate the semi-perimeter: $s = \frac{13 + 14 + 15}{2} = 21$ cm
2. Apply Heron’s Formula: Area = $\sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056} = 84$ cm²

**Using Heron’s Formula in Composite Figures**

Heron’s formula is not just limited to individual triangles; it’s a powerful tool when dealing with composite figures (shapes formed by combining simpler shapes).

Definitions

A composite figure is a shape formed by combining two or more basic geometric shapes, such as triangles, rectangles, squares, etc.

Core Principles

The key principle is to decompose the composite figure into simpler shapes, primarily triangles. Once you have the triangles, you can apply Heron’s formula (or other area formulas for known shapes like rectangles) to find the area of each individual component. Finally, sum the areas of the components to find the total area of the composite figure.

Steps

1. Decomposition: Divide the composite figure into triangles and other basic shapes (like rectangles, squares etc.)
2. Measure Side Lengths: Determine the lengths of all sides of the triangles formed. If side lengths are not directly given, you might need to use other geometric principles (like the Pythagorean theorem).
3. Apply Heron’s Formula (if needed): For each triangle, calculate the area using Heron’s formula (if you have all three side lengths). If it is another shape, apply respective formulas.
4. Calculate the Area: Calculate the area of each shape.
5. Sum the Areas: Add up the areas of all the shapes to find the total area of the composite figure.

Examples

Consider a quadrilateral composed of two triangles.

1. Triangle 1: Sides a = 3 cm, b = 4 cm, c = 5 cm. $s = (3+4+5)/2 = 6$. Area = $\sqrt{6(6-3)(6-4)(6-5)} = \sqrt{36} = 6$ cm²
2. Triangle 2: Sides a = 5 cm, b = 5 cm, c = 6 cm. $s = (5+5+6)/2 = 8$. Area = $\sqrt{8(8-5)(8-5)(8-6)} = \sqrt{144} = 12$ cm²
3. Total Area: 6 cm² + 12 cm² = 18 cm²