Heron's formula is named after Hero of Alexendria, a Greek Engineer and Mathematician in 10 - 70 AD. You can use this formula to find the area of a triangle using the 3 side lengths.

Therefore, you do not have to rely on the formula for area that uses base and height. Diagram 1 below illustrates the general formula where S represents the semi-perimeter of the triangle.

semi-perimeter is just the perimeter divided by 2 : $$ \frac{perimeter}{2} $$ .

Since Heron's formula relates the side lengths, perimeter and area of a triangle, you might need to answer more challenging question types like the following example.

Example 2

Given $$ \triangle ABC $$, with an area of $$ 8.94 $$ square units, a perimeter of $$ 16 $$ units and side lengths $$AB = 3 $$ and $$ CA = 7 $$, what is $$ \red { BC }$$ ?

Step 1

Calculate the semi perimeter, S.

$
S =\frac{ perimeter}{2}
\\
S = \frac{16}{2}
\\
S = 8
$

Step 2

Substitute known values into the formula . Let $$ \red x = \red{ BC} $$

If the perimeter of $$ \triangle ABC $$ is $$32$$ units, its area is $$ 35.8 $$ units squared, and $$ AB= 14 $$ and $$ BC = 12 $$, what is the length of the third side, side $$ \red {CA} $$ ?

If the perimeter of a triangle is 26 units, its area is 18.7 units squared, and the lengths of AB = 12 and BC = 4, what is the length of the third side, side CA?