What is the area (in square centimetres) of an equilateral triangle with side length 6 cm?

Introduction / Context:
This problem tests the standard formula for the area of an equilateral triangle. Since all sides and angles are equal, the area can be computed quickly using a known expression rather than decomposing it into right triangles every time. This is a fundamental result in geometry and frequently appears in aptitude questions.

Given Data / Assumptions:

• The triangle is equilateral, so all sides are equal.
• Side length a = 6 cm.
• We need the area in square centimetres.
• No additional constructions or special angles are required beyond equilateral triangle properties.
• We may use the standard area formula for equilateral triangles.

Concept / Approach:
The area A of an equilateral triangle with side length a is given by A = (√3 / 4) * a^2. This formula can be derived by dropping an altitude and using Pythagoras theorem, but in exam situations it is usually taken as known. Substituting the given side length into this formula will yield the required area in simplest radical form.

Step-by-Step Solution:
Side length a = 6 cm.Area formula for an equilateral triangle: A = (√3 / 4) * a^2.Compute a^2 = 6^2 = 36.Substitute into the formula: A = (√3 / 4) * 36.Simplify: 36 / 4 = 9, so A = 9√3 square centimetres.

Verification / Alternative check:
Alternatively, drop an altitude from one vertex, which splits the triangle into two right triangles with hypotenuse 6 and base 3. Using Pythagoras theorem, altitude h satisfies 6^2 = 3^2 + h^2, so h^2 = 36 − 9 = 27 and h = 3√3. The area is then (1 / 2) * base * height = (1 / 2) * 6 * 3√3 = 9√3, matching the earlier calculation.

Why Other Options Are Wrong:
Option 36√3 is four times the correct area and corresponds to forgetting the factor 1 / 4 in the formula. Option 9 is missing the √3 factor and thus underestimates the area. Option 36 is equivalent to 6^2 and ignores the triangular shape entirely. Option 18√3 is twice the correct area and might result from using an incorrect base or height in the right triangle approach.

Common Pitfalls:
Some students mix up the formula for the area of a triangle with that of a square or rectangle. Others write (√3 / 2) * a^2 instead of (√3 / 4) * a^2. Another common mistake is to miscompute the altitude by taking it as half the side length rather than using Pythagoras theorem.

Final Answer:
The area of the equilateral triangle is 9√3 square centimetres.