Class 9 Mathematic Heron's Formula All Formula & Concept Notes | Formula Sheet

Heron's Formula is used to find the area of a triangle when you know the lengths of all three sides. It's particularly useful when the height of the triangle is not known. Here’s a detailed breakdown:

Heron's Formula

Concept

For a triangle with sides of lengths $a$, $b$, and $c$, Heron's Formula allows you to calculate the area using only these side lengths.

Steps to Use Heron's Formula

1. Calculate the Semi-Perimeter: The semi-perimeter ($s$) of the triangle is half of the perimeter. It is given by:

   $$s = \frac{a + b + c}{2}$$

   where $a$, $b$, and $c$ are the lengths of the sides of the triangle.

2. Apply Heron's Formula: Once you have the semi-perimeter, you can find the area ($A$) of the triangle using:

   $$$$A=s(s−a)(s−b)(s−c)​

   where $s$ is the semi-perimeter, and $a$, $b$, and $c$ are the lengths of the sides.

Example

Let’s go through an example to make it clear.

Example:

Suppose we have a triangle with sides of lengths 7, 8, and 9 units.

1. Calculate the Semi-Perimeter:

   $$s = \frac{7 + 8 + 9}{2} = 12$$

2. Apply Heron's Formula:

   $$A = \sqrt{12 \cdot (12 - 7) \cdot (12 - 8) \cdot (12 - 9)}$$ $$A = \sqrt{12 \cdot 5 \cdot 4 \cdot 3}$$ $$A = \sqrt{720} \approx 26.83 \text{ square units}$$
   

Important Points to Remember

• Triangle Inequality Theorem: For the sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This must be satisfied before applying Heron's Formula.

• Applicability: Heron's Formula works for any triangle (not necessarily right, acute, or obtuse). It’s especially handy when dealing with non-right triangles where you don’t have the height.

• Accuracy: Make sure to use a calculator to get an accurate square root when working with actual numbers.