# A triangle has sides with lengths: 7, 6, and 8. How do you find the area of the triangle using Heron's formula?

Dec 21, 2015

Substitute the lengths into the formula and calculate.

#### Explanation:

Heron's formula states that the area of a triangle whose sides have lengths a, b, and c is
$A = \left\{\setminus \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}\right\}$
where s is the semiperimeter of the triangle; that is,
$s = \left\{\setminus \frac{a + b + c}{2}\right\}$
In this example $s = \frac{7 + 6 + 8}{2} = \frac{21}{2}$
$s - a = \frac{21}{2} - 7 = \frac{21 - 7 \cdot 2}{2} = \frac{7}{2}$
$s - b = \frac{21}{2} - 6 = \frac{21 - 12}{2} - \frac{9}{2}$
$s - c = \frac{21}{2} - 8 = \frac{5}{2}$
$A = \sqrt{\frac{21}{2} \cdot \frac{7}{2} \cdot \frac{9}{2} \cdot \frac{5}{2}}$
$A = \sqrt{\frac{21 \cdot 7 \cdot 9 \cdot 5}{16}}$
$A = \frac{\sqrt{3 \cdot 7 \cdot 7 \cdot 3 \cdot 3 \cdot 5}}{4}$
$A = 7 \cdot 3 \cdot \frac{\sqrt{3 \cdot 5}}{4}$
$A = 21 \cdot \frac{\sqrt{15}}{4}$