The area of an equilateral triangle is given as 36√3 square centimetres. Using the standard area formula for an equilateral triangle, what is the length of each side of the triangle in centimetres?

Introduction / Context:
Here we are given the area of an equilateral triangle and asked to determine the side length. This reverses the usual direct application of the area formula and tests your ability to solve the formula for the side when the area is known, a common style in quantitative aptitude exams.

Given Data / Assumptions:

• The triangle is equilateral.
• Area A = 36√3 sq cm.
• We must find the side length a in cm.
• Standard Euclidean geometry and exact surd representation are assumed.

Concept / Approach:
For an equilateral triangle with side a, its area is: A = (√3 / 4) * a^2 We are given A, so we solve this equation for a. That involves isolating a^2 and taking the positive square root, since side lengths are positive.

Step-by-Step Solution:
Start with A = (√3 / 4) * a^2. We are told A = 36√3. So 36√3 = (√3 / 4) * a^2. Multiply both sides by 4 to remove the denominator: 4 * 36√3 = √3 * a^2. This gives 144√3 = √3 * a^2. Divide both sides by √3: a^2 = 144. Take square root: a = √144 = 12 cm.

Verification / Alternative check:
Substitute a = 12 back into the area formula. Then A = (√3 / 4) * 12^2 = (√3 / 4) * 144 = 36√3 sq cm. This matches the given area, so the side length 12 cm is confirmed as correct.

Why Other Options Are Wrong:
6 cm would give area (√3 / 4) * 36 = 9√3 sq cm, too small. 18 cm would give (√3 / 4) * 324 = 81√3 sq cm, which is more than double the required area. 24 cm produces an even larger area, far from 36√3.

Common Pitfalls:
A common error is to forget the factor 1/4 in the formula or to handle the surd √3 incorrectly when solving for a. Some candidates mistakenly take a to be the square root of 36, instead of correctly isolating a^2. Always rearrange the formula step by step to avoid algebraic slips.

Final Answer:
The side length of the equilateral triangle is 12 cm.