Heron's Formula Calculator

How to Calculate Triangle Area using Heron's Formula

Heron's formula is a method for calculating the area of a triangle when you know the lengths of all three sides. This formula is particularly useful when you don't have information about the triangle's height or angles.

Heron's Formula

Let \( a \), \( b \), and \( c \) be the side lengths of a triangle. Then the area \( A \) of the triangle is given by:

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

Where \( s \) is the semi-perimeter of the triangle:

\[ s = \frac{a + b + c}{2} \]

Calculation Steps

1. Measure or identify the lengths of all three sides of the triangle: \( a \), \( b \), and \( c \).
2. Calculate the semi-perimeter \( s \) using the formula: \( s = \frac{a + b + c}{2} \)
3. Apply Heron's formula: \( A = \sqrt{s(s-a)(s-b)(s-c)} \)
4. Simplify and calculate the final result.

Example Calculation

Let's calculate the area of a triangle with sides \( a = 3 \), \( b = 4 \), and \( c = 5 \):

1. Calculate the semi-perimeter: \[ s = \frac{a + b + c}{2} = \frac{3 + 4 + 5}{2} = \frac{12}{2} = 6 \]
2. Apply Heron's formula: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] \[ A = \sqrt{6(6-3)(6-4)(6-5)} \] \[ A = \sqrt{6 \cdot 3 \cdot 2 \cdot 1} \] \[ A = \sqrt{36} = 6 \]

Therefore, the area of the triangle is 6 square units.

Visual Representation

Here's a visual representation of a triangle with sides 3, 4, and 5:

This diagram illustrates a right-angled triangle (3-4-5 triangle), which is a special case where Heron's formula can be applied, although it works for any triangle.