AUTHOR - Kairav Nandi
“If the area of an equilateral triangle is 36√3 sq cm, find its perimeter.”
This is a question that came in our mathematics exam.
Now, we know that first, we will find out the side of the triangle first, after which, we have to multiply it with 3 to get the perimeter. One would do it that way itself. But then, a thought came- Can we find out the perimeter from the area directly and bypass calculating the length of the side? Turns out, we actually can. The proof regarding it has been provided below.
Let the area of the equilateral triangle be ‘s’, the length of the side be ‘a’, and the perimeter be ‘p’.
Now, we already know that
s = (a²√3)/4 sq units
Now, let us find out the value of ‘a’ from here.
Therefore, 4s = a²√3 [ transferring ¼ to LHS]
=> 4s/√3 = a² [transferring √3 to LHS]
=> a = √(4s/√3)
= √4√s/√(√3) [√(xy) = √x√y] [√(x/y) = √x/√y]
=> a = 2√s/3^1/4
Now that we have the value of ‘a’, we can find out the perimeter by multiplying ‘a’ with 3.
Therefore,
p = 3a
=> p = 3(2√s/31/4)
=> p = 2√s × 3/3^1/4
= 2√s × 3^1 - 1/4 [xn/xm = xn-m]
= 2√s × 3^4/4 - ¼
=> p = 2√s × 33/4 units
Now let us verify this formula using the question that was asked-
Area given (s) = 36√3 sq cm
Therefore, perimeter = 2√s × 3^3/4
= 2√(36√3) × 3^3/4
= 2√36 × (√3) × 3^3/4
= 2 × 6 × 31/4 × 33/4
= 12 × 31/4+3/4 [xm×xn=xm+n]
= 12 × 34/4 = 12 × 3 = 36 cm
Alternatively, we can find out the perimeter from the length of the side-
Area given (s) = 36√3 sq cm
Formula for finding out area in an Equilateral Triangle = (a²√3)/4, where 'a' is the length of the side.
Therefore, (a²√3)/4 = 36√3
=> a²√3 = 36√3 × 4 [transferring ¼ to RHS]
=> a² = 36 × 4 [cancelling √3 from both sides]
=> a = √(36 × 4)
= √36 × √4 [√(xy) = √x × √y]
= 6 × 2 = 12 cm
Therefore, Perimeter = 3 × 12 = 36 cm Hence, the formula has been proved and verified.
Thank You for reading this.
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