Question

The perimeter of a particular equilateral triangle is numerically equal to the area of the triangle. Find the perimeter of the triangle.
A. \[3\]
B. \[4\]
C. \[4\sqrt 3 \]
D. \[12\sqrt 3 \]
E. \[18\sqrt 3 \]

Answer 1

Hint: To solve this question we will use the formula of perimeter of a triangle and area of an equilateral triangle. An equilateral triangle is one whose all three sides are equal.

Formula used: Perimeter of a triangle\[ = side + side + side\]
And for an equilateral triangle, Perimeter=\[3 \times side\]
Area of an equilateral triangle \[A = \dfrac{{\sqrt 3 }}{4} \times {(side)^2}\] [we can also use Pythagoras theorem to calculate area of HERON’S formula to calculate, but it is best to memorise this formula of area to save our time.]

Complete step by step solution: Let the side of the equilateral triangle be ‘a’ unit.
So, first calculate the perimeter (P) of this triangle,
i.e. \[P = 3 \times a = 3a\]
Now calculate Area (A) of this triangle,
i.e. \[A = \dfrac{{\sqrt 3 }}{4} \times {a^2}\]
Now it is given in the question that the perimeter and area of the equilateral triangle is numerically the same, So we equate both A and P.
i.e. \[3a = \dfrac{{\sqrt 3 }}{4} \times {a^2}\]
Or \[\dfrac{{{a^2}}}{a} = \dfrac{3}{{\dfrac{{\sqrt 3 }}{4}}}\]
Or \[a = \dfrac{{4 \times 3}}{{\sqrt 3 }}\]
Or \[a = \dfrac{{4 \times 3 \times \sqrt 3 }}{{\sqrt 3 \times \sqrt 3 }}\] [We multiply numerator and denominator by square root term i.e. which is known as rationalizing of denominator]
Or \[a = \dfrac{{4 \times 3 \times \sqrt 3 }}{3}\]
Or \[a = 4\sqrt 3 \]
Hence, the side of the equilateral triangle is \[4\sqrt 3 \].
In last, put this value of a in the formula of perimeter P, we get:
\[P = 3a = 3 \times 4\sqrt 3 = 12\sqrt 3 \]

Hence option D is our correct answer.

Note: Whenever a square root term comes in the denominator of any equation, rationalise it always to match your proper answer. Students must be careful about applying perimeter and area formula for triangle type. In question the triangle is an equilateral triangle. Also memorise the formula of area of an equilateral triangle to save your time. Apply the condition given in the question properly. Most students just calculate the value of a in this question and mark the answer, but here we have to calculate perimeter, so read the statement carefully.