What is the length of the side of an equilateral triangle if its area is given to be 36√3 square centimetres?

Introduction / Context:
This question tests your understanding of the formula for the area of an equilateral triangle and your ability to manipulate square roots and algebraic expressions. Equilateral triangles have equal sides and equal angles, which leads to a special and widely used area formula. Problems like this commonly appear in aptitude exams to check whether you can move smoothly from area back to side length.

Given Data / Assumptions:

• The triangle is equilateral, so all sides are equal.
• Area of the triangle is A = 36√3 square centimetres.
• We must find the side length a of the triangle in centimetres.
• We will use the standard formula for the area of an equilateral triangle.

Concept / Approach:
For an equilateral triangle with side length a, the area is given by:
A = (√3 / 4) * a^2. We know A and must solve this equation for a. This involves basic algebra: isolating a^2 and then taking the square root to find a. Careful handling of the square root factors ensures an exact and clean answer.

Step-by-Step Solution:
Step 1: Write the area formula: A = (√3 / 4) * a^2. Step 2: Substitute the given area: 36√3 = (√3 / 4) * a^2. Step 3: Multiply both sides by 4 to eliminate the denominator: 4 * 36√3 = √3 * a^2. Step 4: Compute 4 * 36√3 = 144√3. Step 5: The equation becomes 144√3 = √3 * a^2. Step 6: Divide both sides by √3: a^2 = 144. Step 7: Take the positive square root (side length is positive): a = √144 = 12. Step 8: Therefore, the side length of the equilateral triangle is 12 cm.

Verification / Alternative check:
Check by substituting a = 12 back into the area formula. Compute a^2 = 12^2 = 144. Then area A = (√3 / 4) * 144. Simplify: 144 / 4 = 36, so A = 36√3. This matches the area given in the problem exactly, confirming that the side length was calculated correctly and that no algebraic steps were missed or mishandled.

Why Other Options Are Wrong:
If a = 6 cm, the area would be (√3 / 4) * 36 = 9√3, which is too small. If a = 24 cm, the area would be (√3 / 4) * 576 = 144√3, which is much larger than 36√3. If a = 18 cm, the area becomes (√3 / 4) * 324 = 81√3, also too large. The value 9 cm gives an area (√3 / 4) * 81 = 20.25√3, which still does not match. Only a side length of 12 cm yields area 36√3.

Common Pitfalls:
A common problem is misremembering the formula and using (1 / 2) * base * height without computing the height correctly. Another error is incorrectly handling the factor √3 / 4 when solving for a^2, especially when multiplying both sides by 4 or dividing by √3. Some students also make arithmetic mistakes when squaring or taking square roots, such as forgetting that √144 is 12 and not another value. Writing each step clearly helps avoid these errors.

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