Question

How do find the area of a triangle with 3 sides given?

Answer 1

Hint: In the given question, we have been asked how we can calculate the area of any triangle with the measure of all its sides given. Here, we have not been given what type of triangle is it – isosceles, scalene, equilateral or right-angled. So, we are going to have to write the formula which can solve for the area of any given triangle.

Formula Used:
We are going to use the Heron’s formula in the question, which is:
If \[a,b,c\] are the three sides of the triangle and \[s = \dfrac{{a + b + c}}{2}\] be the semi-perimeter, then
\[Area = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \]

Complete step-by-step answer:
In the given question, we have to find the area of any triangle with the length of all the sides given.
We do not know the type of triangle, so we are going to have to write the formula which can solve for any triangle’s area.
Let the sides of the triangle given be \[a,b,c\].
Let \[s\] be the semi-perimeter of triangle, \[s = \dfrac{{a + b + c}}{2}\]
Then, we can solve for the area of this triangle by applying Heron’s formula.
\[Area = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \]

Additional Information:
If we have an equilateral triangle particularly mentioned, then we can use the formula \[\dfrac{{\sqrt 3 }}{4}{a^2}\]. Now, we can check for the formula through Heron’s formula.
The sides of the triangle are all \[a\] and hence, \[s = \dfrac{{3a}}{2}\]
Thus, \[Area = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \]
or, \[A = \sqrt {\dfrac{{3a}}{2}\left( {\dfrac{{3a}}{2} - a} \right)\left( {\dfrac{{3a}}{2} - a} \right)\left( {\dfrac{{3a}}{2} - a} \right)} \]
\[A = \sqrt {\dfrac{{3a}}{2} \times \dfrac{a}{2} \times \dfrac{a}{2} \times \dfrac{a}{2}} = \sqrt {3 \times {{\left( {\dfrac{a}{2}} \right)}^4}} = \sqrt 3 \times \dfrac{{{a^2}}}{4} = \dfrac{{\sqrt 3 }}{4}{a^2}\]
Hence, verified.

Note: If the triangle is given to be an equilateral, we could have simply applied the formula – \[\dfrac{{\sqrt 3 }}{4}{a^2}\], with \[a\] equal to measure of each side. If the triangle is right-angled, we can apply the formula – \[\dfrac{1}{2} \times base \times height\]. But other than that, we are going to have to apply Heron’s formula to solve it.