Heron's Formula: Calculating Triangle Area

Heron's Formula is a powerful tool used to calculate the area of a triangle when the lengths of all three sides are known. Unlike other methods that require the height of the triangle, Heron's Formula only needs the side lengths, making it incredibly versatile for various geometric problems.

The Formula

Let a, b, and c be the lengths of the three sides of a triangle. First, we need to calculate the semi-perimeter (s) of the triangle, which is half of its perimeter:

s = (a + b + c) / 2

Once the semi-perimeter is found, the area (A) of the triangle can be calculated using Heron's Formula:

A = √(s * (s - a) * (s - b) * (s - c))

Steps to Calculate Area Using Heron's Formula

1. Identify the side lengths: Determine the values for a, b, and c.
2. Calculate the semi-perimeter (s): Add the three side lengths and divide by 2.
3. Subtract each side length from the semi-perimeter: Calculate (s - a), (s - b), and (s - c).
4. Multiply the semi-perimeter by the three differences: Calculate s * (s - a) * (s - b) * (s - c).
5. Take the square root: The square root of the product from step 4 is the area of the triangle.

Step-by-Step Example

Let's find the area of a triangle with side lengths a = 7 cm, b = 10 cm, and c = 13 cm.

1. Side lengths:
   a = 7 cm
   b = 10 cm
   c = 13 cm
2. Calculate the semi-perimeter (s):
   s = (7 + 10 + 13) / 2
   s = 30 / 2
   s = 15 cm
3. Subtract each side length from the semi-perimeter:
   (s - a) = 15 - 7 = 8 cm
   (s - b) = 15 - 10 = 5 cm
   (s - c) = 15 - 13 = 2 cm
4. Multiply s by the three differences:
   15 * 8 * 5 * 2 = 1200
5. Take the square root:
   A = √1200
   A ≈ 34.64 cm²

Therefore, the area of the triangle is approximately 34.64 cm².