Question

The area of an equilateral triangle is numerically equal to its perimeter. Find the length of its side.

Answer 1

Hint: We will equate the formulae of the area and perimeter of the equilateral triangle assuming the side of the given equilateral triangle as a. Then, we will obtain an equation in a and we will solve it for the value of a and it will be the required length.

Complete step-by-step answer:
We are given that the area and the perimeter of an equilateral triangle are numerically equal.
We are required to calculate the value of the length of the side of the given equilateral triangle.
Let the side of the equilateral triangle be a . Since it is an equilateral triangle, the length of all sides of the triangle will be equal.

Let ABC be the equilateral triangle with sides a is ${\text{a cm}}$.
The perimeter of the triangle is the sum of all sides of the triangle. So, the perimeter of the equilateral triangle with side a is equal to $(a + a + a)cm = 3a{\text{ cm}}$.
The formula of area of the equilateral triangle with side is: $\dfrac{{\sqrt 3 }}{4}{a^2}c{m^2}$
According to the question, the area of the equilateral triangle is equal to the perimeter of the given equilateral triangle.
Writing it in terms of an equation by putting the values of the area and the perimeter of the equilateral triangle, we get
$ \Rightarrow \left( {\dfrac{{\sqrt 3 }}{4}{a^2}} \right) = 3a$
Solving this equation for the value of a, we get
$ \Rightarrow a = \dfrac{{3\left( 4 \right)}}{{\sqrt 3 }} = \dfrac{{\sqrt 3 \times \sqrt 3 \times 4}}{{\sqrt 3 }} = 4\sqrt 3 $
So, the length of the side of the given equilateral triangle is $4\sqrt 3 cm$.
Note: We may get confused while solving for a after writing the equation based on the data given (i.e., the area and the perimeter of an equilateral triangle are equal). This question is based on implementation of the formulae known directly in the solution. We can also put the value of $\sqrt 3 = 1.732$ in the answer obtained to simplify it further. The length will be $4 \times 1.732 = 6.928cm$ or $7cm$(approximately).