Question

Area of an equilateral triangle of side ‘a’ units can be calculated by using the formula:
A. \[\sqrt{{{s}^{2}}{{\left( s-a \right)}^{2}}}\]
B. \[\left( s-a \right)\sqrt{{{s}^{2}}\left( s-a \right)}\]
C. \[\sqrt{s{{\left( s-a \right)}^{2}}}\]
D. \[\left( s-a \right)\sqrt{s\left( s-a \right)}\]

Answer 1

- Hint:- Use Heron’s formula. As it is an equilateral triangle all the sides are equal. Substitute the side as ‘a’ in the equation and simplify it. Thus find the area.

Complete step-by-step answer: -
We can find the area of a triangle using Heron’s formula, when the length of 3 sides of the triangle are given then the area of the triangle can be found using Heron’s formula.
\[\therefore \]The area of the triangle is calculated using the equation,
\[A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}.......(1)\]
where a, b, c are the side lengths of the triangle and s is half of the perimeter.
The value of s is found using the formula, \[s=\dfrac{a+b+c}{2}\].
Equation (1) is for a triangle with 3 different sides, i.e. the value of the sides is different. We need to find the area of an equilateral triangle with side ‘a’, which means that all sides are equal and of value ‘a’.
We know that in an equilateral triangle, all sides are equal. Thus here, a = b = c.
Thus in equation (1) put a = b = c for an equilateral triangle. Thus the area of the equilateral triangle becomes,
\[\begin{align}
  & A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}=\sqrt{s\left( s-a \right)\left( s-a \right)\left( s-a \right)}=\sqrt{s{{\left( s-a \right)}^{3}}} \\
 & =\sqrt{s\left( s-a \right){{\left( s-a \right)}^{2}}}=\left( s-a \right)\sqrt{s\left( s-a \right)} \\
\end{align}\]
Thus we got the area of the equilateral triangle as, \[A=\left( s-a \right)\sqrt{s\left( s-a \right)}\].
Thus we got the required area of the equilateral triangle.
Option D is the correct answer.

Note:- From the options provided, you should be able to assume that Heron’s formula is used because by other methods of calculating area, we can’t get the required answer. The other methods of finding area includes, \[A={}^{1}/{}_{2}bh\] and then by using distance formula if coordinates of the vertex of the triangle is provided.