Question

How do you use Heron’s formula to determine the area of a triangle with sides of that are 4,7 and 8 units in length?

Answer 1

Hint: Heron’s formula is used to determine the area of a triangle when three sides are given .
We should first calculate $s$which is half of the triangle’s perimeter
$s = \dfrac{{a + b + c}}{2}$
The area is calculated as,
$A = \sqrt {s(s - a)(s - b)(s - c)} $

Complete step by step answer:
The three sides of the triangle are $4,7$and$8$.
Hence,$a = 4,b = 7,c = 8$
Now , calculating the value of $s$ we get,
$
\Rightarrow s = \dfrac{{a + b + c}}{2} \\
\Rightarrow s = \dfrac{{4 + 7 + 8}}{2} \\
\Rightarrow s = \dfrac{{19}}{2} \\
    \\
 $
We use this $s$value to calculate the area
$
\Rightarrow A = \sqrt {s(s - a)(s - b)(s - c)} \\
\Rightarrow A = \sqrt {\dfrac{{19}}{2}\left( {\dfrac{{19}}{2} - 4} \right)\left( {\dfrac{{19}}{2} - 7} \right)\left( {\dfrac{{19}}{2} - 8} \right)} \\
\Rightarrow A = \sqrt {\dfrac{{19}}{2}\left( {\dfrac{{19 - 8}}{2}} \right)\left( {\dfrac{{19 - 14}}{2}} \right)\left( {\dfrac{{19 - 16}}{2}} \right)} \\
\Rightarrow A = \sqrt {\dfrac{{19}}{2}\left( {\dfrac{{11}}{2}} \right)\left( {\dfrac{5}{2}} \right)\left( {\dfrac{3}{2}} \right)} \\
\Rightarrow A = \dfrac{1}{4}\sqrt {3135} \\
\Rightarrow A = \dfrac{1}{4} \times 55.99 \\
\Rightarrow A = 13.99 \simeq 14squnits \\
 $

Hence, the area of the triangle is $14 squnits$

Note: Heron’s formula is applicable only if all the three sides of the triangle are given. It is not suitable for right angled triangles because the process of solving becomes very tedious. Instead we can directly find the area using the formula $A = \dfrac{1}{2} \times b \times h$, where $b$ is the base and $h$ is the height of the triangle.