Heron’s Formula is a remarkable method for calculating the area of a triangle when the lengths of all three sides are known. This formula is particularly useful when the height of the triangle is not readily available.

Let the lengths of the sides of the triangle be denoted as $a$, $b$, and $c$. The first step in using Heron’s Formula is to compute the semi-perimeter $s$ of the triangle. The semi-perimeter is given by:

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

Once the semi-perimeter is determined, the area $A$ of the triangle can be calculated using the following formula:

$A = \sqrt{s(s - a)(s - b)(s - c)}$

In this formula:

• $s$ is the semi-perimeter.
• $(s - a)$, $(s - b)$, and $(s - c)$ are the differences between the semi-perimeter and each of the sides of the triangle.

Example:

Consider a triangle with sides $a = 7$ units, $b = 8$ units, and $c = 9$ units. We can calculate its area using Heron’s Formula as follows:

1. First, compute the semi-perimeter:

$s = \frac{7 + 8 + 9}{2} = \frac{24}{2} = 12$ units

2. Next, substitute $s$ into Heron’s Formula:

$A = \sqrt{12(12 - 7)(12 - 8)(12 - 9)}$

3. Calculate each term:

$A = \sqrt{12 \times 5 \times 4 \times 3}$

$= \sqrt{720}$

$= 12\sqrt{5}$ square units

This result shows that the area of the triangle is $12\sqrt{5}$ square units.

Important Properties:

• Heron’s Formula can be applied to any triangle, whether it is acute, obtuse, or right-angled.
• It simplifies calculations when the height of the triangle is not known.

Applications:

Heron’s Formula is used in various real-life applications, such as:

• Construction, to determine land area.
• Land surveying, to calculate areas of triangular plots.

Limitations:

• While Heron’s Formula is very useful, it requires knowledge of all three sides of the triangle.
• It may not be the most efficient method when the height can be easily calculated.