# Heron’s Formula

Definition: For any given triangle, the area can be found with Heron’s Formula, where s is half the perimeter:

$\dpi{150} \bg_white \small \\Area\;of\;\Delta ABC = \sqrt{ s\cdot (s-a) \cdot (s-b) \cdot (s-c)}\\ \\\\\textup{where,} \;\;\;s = \;\;\frac{perimeter}{2}\;\; =\;\; \frac{a+b+c}{2}$

Example: Given a triangle with side lengths 4, 5 and 6 find the area.

Solution:

Step 1: Find s, half the perimeter.

s = (3 + 4 + 5)/2 = 15/2 = 7.5

Step 2: Plug all values into the formula, keeping in mind that a, b and c are the side lengths.

$\dpi{150} \bg_white \small \\A = \sqrt{ s\cdot (s-a) \cdot (s-b) \cdot (s-c)}\\ \\A = \sqrt{7.5\cdot (7.5-4) \cdot (7.5-5) \cdot (7.5-6)}\\$

Step 3: Simplify to solve for the area.

$\dpi{150} \bg_white \small \\A = \sqrt{7.5\cdot (7.5-4) \cdot (7.5-5) \cdot (7.5-6)}\\ \\A = \sqrt{7.5\cdot 3.5 \cdot 2.5 \cdot 1.5}\\ \\A = \sqrt{98.4375}\; \approx \;9.92 \;units^2$

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