# How do you use Heron's formula to find the area of a triangle with sides of lengths 29 , 25 , and 12 ?

Jun 20, 2016

Area of triangle is $148.92$

#### Explanation:

If the sides of a triangle are $a$, $b$ and $c$, then according to Heron's formula, the area of the triangle is given by the formula

$\Delta = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$, where $s = \frac{1}{2} \left(a + b + c\right)$

Now given the sides of a triangle as $29$, $25$ and $12$

$s = \frac{1}{2} \times \left(29 + 25 + 12\right) = \frac{1}{2} \times 66 = 33$ and

$\Delta = \sqrt{33 \left(33 - 29\right) \left(33 - 25\right) \left(33 - 12\right)}$

= $\sqrt{33 \times 4 \times 8 \times 21}$

= $\sqrt{3 \times 11 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 7}$

= $3 \times 2 \times 2 \sqrt{11 \times 2 \times 7} = 12 \sqrt{154} = 12 \times 12.41 = 148.92$