# How do you use Heron's formula to determine the area of a triangle with sides of that are 25, 28, and 21 units in length?

Dec 29, 2015

$24 \sqrt{111}$ ${\text{units}}^{2}$

#### Explanation:

Heron's formula:

$A = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

When $s$, the semiperimeter, is equal to

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

when $a , b , c$ are the sides of the triangle.

Find $s$ first:

$s = \frac{25 + 28 + 21}{2} = 37$

Thus,

$A = \sqrt{37 \left(37 - 25\right) \left(37 - 28\right) \left(37 - 21\right)}$

$A = \sqrt{37 \times 12 \times 9 \times 16}$

$A = \sqrt{37} \times \sqrt{{2}^{2} \times 3} \times \sqrt{{3}^{2}} \times \sqrt{{4}^{2}}$

$A = \left(2 \times 3 \times 4\right) \sqrt{37 \times 3}$

$A = 24 \sqrt{111} \approx 252.8557$