The area of an equilateral triangle is 64√3 sq cm. What is the length of each side of this triangle in centimetres?

Introduction / Context:
This problem reverses the standard equilateral triangle area formula. Instead of being given the side and asked for the area, you are given the area and asked to find the side length. This reverse application is very common in competitive exams and ensures that you really understand the formula and can manipulate it algebraically, not just plug numbers in one direction.

Given Data / Assumptions:

• The triangle is equilateral, so all sides are equal and all angles are 60 degrees.
• The area A of the triangle is 64√3 sq cm.
• We are required to find the side length in centimetres.
• Standard equilateral triangle properties and formulas apply.
• No rounding is necessary because the answer is an exact integer.

Concept / Approach:
The area of an equilateral triangle with side length a is A = (√3 / 4) * a^2. Given A, we solve this equation for a. First we isolate a^2 by dividing both sides by (√3 / 4), and then we take the positive square root because side lengths are positive. Once we have a, we can verify by substituting back into the formula to check that the area matches 64√3 sq cm.

Step-by-Step Solution:
Given A = 64√3 sq cm and A = (√3 / 4) * a^2.Set (√3 / 4) * a^2 = 64√3.Divide both sides by √3: (1 / 4) * a^2 = 64.Multiply both sides by 4: a^2 = 256.Take the positive square root: a = 16 cm.

Verification / Alternative check:
To verify, compute the area again using a = 16 cm. Using A = (√3 / 4) * a^2, we get A = (√3 / 4) * 16^2 = (√3 / 4) * 256. Simplifying gives A = 64√3 sq cm, which matches the given area exactly. This confirms that the side length 16 cm is correct and consistent with the formula and the provided data.

Why Other Options Are Wrong:
Option 8 cm would give area (√3 / 4) * 8^2 = 16√3, which is much smaller than 64√3. Option 16√3 cm is far too large and would lead to area that grows with the square of that expression, not 64√3. Option 8√3 cm also produces a different area. Option 12 cm gives area (√3 / 4) * 144 = 36√3, which again does not match 64√3. Only 16 cm satisfies the equation derived from the given area.

Common Pitfalls:
Some students forget to divide by √3 and instead cancel incorrectly, leading to wrong values for a^2. Others might misapply the formula and use A = (1 / 2) * a^2, which is not correct for an equilateral triangle. Another typical mistake is to take the square root of 64 only and ignore the factor of 4 from the denominator. Working step by step and treating the equation carefully avoids these errors.

Final Answer:
The length of each side of the equilateral triangle is 16 cm.