How do you use Heron's formula to determine the area of a triangle with sides of that are 8, 3, and 10 units in length?

Sep 10, 2016

Area of triangle is $9.922$

Explanation:

According to Heron's formula, if the sides of a triangle are $a$, $b$ and $c$, then the area of the triangle $\Delta$ is given by the formula

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

Here we have three sides of a triangle as $8$, $3$ and $10$.

Hence $s = \frac{1}{2} \left(8 + 3 + 10\right) = \frac{1}{2} \times 21 = \frac{21}{2}$

and $\Delta = \sqrt{\frac{21}{2} \left(\frac{21}{2} - 8\right) \left(\frac{21}{2} - 3\right) \left(\frac{21}{2} - 10\right)}$

= $\sqrt{\frac{21}{2} \times \frac{5}{2} \times \frac{15}{2} \times \frac{1}{2}}$

= $\frac{1}{4} \sqrt{3 \times 7 \times 5 \times 3 \times 5}$

= $\frac{15}{4} \times \sqrt{7}$

= $\frac{15}{4} \times 2.6458$

= $9.922$