Heron's Formula

Overview

Heron's formula calculates the area of a triangle when you know all three side lengths, without needing to find the height.

The Formula

For a triangle with sides $a$, $b$, and $c$:

Step 1: Calculate the Semi-perimeter

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

Step 2: Apply Heron's Formula

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

Alternative Forms

Expanded Form

$\text{Area} = \frac{1}{4}\sqrt{(a + b + c)(-a + b + c)(a - b + c)(a + b - c)}$

Using Side Lengths Only

$\text{Area} = \frac{1}{4}\sqrt{2a^2b^2 + 2b^2c^2 + 2c^2a^2 - a^4 - b^4 - c^4}$

Example Calculation

Problem: Find the area of a triangle with sides 5, 6, and 7.

Solution:

1. Calculate semi-perimeter:

$s = \frac{5 + 6 + 7}{2} = \frac{18}{2} = 9$

2. Calculate each factor:

$s - a = 9 - 5 = 4$

$s - b = 9 - 6 = 3$

$s - c = 9 - 7 = 2$

3. Apply formula:

$\text{Area} = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} = 6\sqrt{6} \approx 14.70 \text{ square units}$

Special Cases

Equilateral Triangle (all sides = $a$)

$s = \frac{3a}{2}$

$\text{Area} = \sqrt{\frac{3a}{2} \cdot \frac{a}{2} \cdot \frac{a}{2} \cdot \frac{a}{2}} = \frac{\sqrt{3}}{4}a^2$

Isosceles Triangle (two sides = $a$, base = $b$)

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

$\text{Area} = \frac{b}{4}\sqrt{4a^2 - b^2}$

Right Triangle (legs $a$, $b$; hypotenuse $c$)

Using Heron's formula will give:

$\text{Area} = \frac{1}{2}ab$

(Same as the standard formula)

Triangle Inequality Check

Before applying Heron's formula, verify the triangle is valid:

$a + b > c$

$b + c > a$

$a + c > b$

If any of these fail, no triangle exists with those side lengths.

When to Use Heron's Formula

• You know all three side lengths
• Finding the height is difficult or impossible
• The triangle is not a right triangle
• You're working with coordinate geometry (use distance formula for sides)

Related Formulas

Area using two sides and included angle

$\text{Area} = \frac{1}{2}ab\sin(C)$

Area using base and height

$\text{Area} = \frac{1}{2}bh$

Circumradius $R$

$\text{Area} = \frac{abc}{4R}$

Inradius $r$

$\text{Area} = rs \quad \text{(where } s \text{ is semi-perimeter)}$

$r = \frac{\text{Area}}{s}$