Heron’s Formula

Heron’s Formula is a mathematical formula used to calculate the area of a triangle when the lengths of all three sides are known. It was named after Hero of Alexandria, a Greek mathematician who first described the formula in his book "Metrica" around 60 AD.

Formula:
The formula for calculating the area of a triangle using Heron’s Formula is as follows:

Area (A) = √(s(s-a)(s-b)(s-c))

Where,
s = (a+b+c)/2
a, b, and c are the lengths of the sides of the triangle.

Explanation:
To understand the formula, let's consider a triangle with sides of length a, b, and c. The formula makes use of the semi-perimeter of the triangle, denoted by s, which is calculated by taking the sum of all three sides and dividing it by 2.

The area of the triangle is then calculated by taking the square root of the product of s and the differences between s and each of the sides (s-a, s-b, and s-c).

Example:
Let's say we have a triangle with sides of length 5 cm, 6 cm, and 7 cm. To calculate the area using Heron’s Formula, we need to first calculate the semi-perimeter, s.

s = (5+6+7)/2 = 18/2 = 9

Next, we substitute the values of s, a, b, and c into the formula:

Area (A) = √(9(9-5)(9-6)(9-7))
= √(9*4*3*2)
= √(216)
≈ 14.7 cm²

Therefore, the area of the triangle is approximately 14.7 square centimeters.

Applications:
Heron’s Formula is particularly useful when the height of the triangle is not known, or when it is not convenient to calculate the height using other methods. It is often used in geometry, engineering, and physics to calculate the area of various shapes.

Overall, Heron’s Formula provides a simple and efficient way to calculate the area of a triangle using only the lengths of its sides.