Heron's Formula

A triangle is ABC is given with the side lengths a, b, c as shown in the figure.

The area of the above triangle is given by Heron's Formula which is

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

where s is semi-perimeter of the triangle.

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

Example:

A scalene triangle has sides of lengths 7 cm, 8 cm, 9 cm. Find area of the triangle.

Solution:

Consider,

a=7, b=8, c=9

$s=\frac{7+8+9}{2}=12$

Now, Substituting all the values in the formula, we get,

$\\Area=\sqrt{12(12-7)(12-8)(12-9)}=\sqrt{12(5)(4)(3)}=12\sqrt5 \\\\Area=12\sqrt5\; sq.cm$

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