Question

The area of an equilateral triangle is \[16\sqrt 3 \,c{m^2}\]. Find its perimeter.

Answer 1

Hint:
Here, equate the area given to the formula for equilateral triangle for finding the side of the triangle. Once sides obtained, substitute the value of the side into the formula for the perimeter of the equilateral triangle.
Formula used: The formula used for the area of the equilateral triangle is \[A = \dfrac{{\sqrt 3 }}{4}{a^2}\]. The formula for the perimeter of the equilateral triangle is \[P = 3a\]. In both the formula, \[a\] is the side of the triangle.

Complete step by step solution:
The area of the equilateral triangle is given which is \[16\sqrt 3 \,c{m^2}\].
We know that the formula of the area of the equilateral triangle is, \[A = \dfrac{{\sqrt 3 }}{4}{a^2}\] ,where \[A\] is the area of equilateral triangle.
 To find the side of the equilateral triangle, equate the area given with the formula for the area of the triangle.
\[\dfrac{{\sqrt 3 }}{4}{a^2} = 16\sqrt 3 \,\]
Now, cancel\[\sqrt 3 \]from both sides.
\[\dfrac{1}{4}{a^2} = 16\,\]
\[{a^2} = 16 \times 4\,\,\]
\[ = 64\]
Now, take the square root on both the sides, Then
\[a = \pm 8\,{\text{cm}}\]
As the length of a triangle can never be negative so we will consider \[a = 8\,cm\] only.
Now use the formula \[P = 3a\] to find the perimeter.
Sustitute \[a = 8\] in the formula.
\[P = 3 \times 8\,cm\]
\[ = 24\,cm\].

Therefore the perimeter is \[24\,cm\].

Note:
The perimeter can also be calculated by adding all three sides of the triangle. Area of can also be a triangle using the formula \[A = \dfrac{1}{2} \times b \times h\], where \[b\] and \[h\] are the base and height of the triangle. In this method we have to find the height of the equilateral triangle using the Pythagoras theorem which is \[{H^2} = {P^2} + {B^2}\] after getting the value of height we can substitute its value as the length of base is already given.
The area of the triangle can also be calculated by using Heron’s formula.