Question

How do you use the Heron’s formula to determine the area of a triangle with sides of that are \[14\], $8$, and $13$ units in length?

Answer 1

Hint: The Heron’s formula for finding out the area of a triangle is given by the expression$\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$, where a, b, and c are the lengths of the sides of the triangle, and s is the semi perimeter of the triangle, which is given by $s=\dfrac{a+b+c}{2}$. According to the above question, we can let $a=14$, $b=8$, and $c=13$ and calculate the value of s. Finally, on substituting these values in the Heron’s formula, we will get the required area of the given triangle.

Complete step-by-step answer:
According to the question, the lengths of the sides of the given triangle are equal to \[14\], $8$, and $13$ units. Let us represent these lengths by a, b and c so that we can write
$\begin{align}
  & \Rightarrow a=14.......\left( i \right) \\
 & \Rightarrow b=8........\left( ii \right) \\
 & \Rightarrow c=13.......\left( iii \right) \\
\end{align}$
From this information, we can draw the given triangle as shown in the below figure.

Now, we know that the Heron’s formula for the area of a triangle is given by the formula
$\Rightarrow A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}.......\left( iv \right)$
Here s is the semi-perimeter of the triangle, and is therefore given by
$\Rightarrow s=\dfrac{a+b+c}{2}$
Substituting the values of a, b, and c from the equations (i), (ii), and (iii) we get
$\begin{align}
  & \Rightarrow s=\dfrac{14+8+13}{2} \\
 & \Rightarrow s=\dfrac{35}{2} \\
 & \Rightarrow s=17.5......\left( v \right) \\
\end{align}$
Now, we substitute the equations (i), (ii), (iii), and (v) into the equation (v) to get
\[\begin{align}
  & \Rightarrow A=\sqrt{17.5\left( 17.5-14 \right)\left( 17.5-8 \right)\left( 17.5-13 \right)} \\
 & \Rightarrow A=\sqrt{17.5\times 3.5\times 9.5\times 4.5} \\
 & \Rightarrow A=51.17 \\
\end{align}\]
Hence, the area of the given triangle is equal to \[51.17\] square units.

Note: We must not misunderstand the semi perimeter s to be the average of the three sides and write it as \[s=\dfrac{a+b+c}{3}\]. Since the perimeter of a closed shape is equal to the sum of all the sides, s will be equal to half the perimeter and given as $s=\dfrac{a+b+c}{2}$.