A regular hexagon (six-sided polygon with all sides and angles equal) has each side of length 6 cm. Using the standard formula, what is the area of this regular hexagon in square centimetres?

Introduction / Context:
This question involves finding the area of a regular hexagon using its side length. A regular hexagon can be divided into six congruent equilateral triangles, which allows us to use the equilateral triangle area formula. Recognising this decomposition is key to solving such problems quickly.

Given Data / Assumptions:

• The polygon is a regular hexagon.
• Each side has length 6 cm.
• We need the total area in sq cm.
• All internal angles are 120°, and all sides are equal.

Concept / Approach:
A regular hexagon of side a can be partitioned into 6 equilateral triangles, each with side a. The area of one equilateral triangle with side a is: A_triangle = (√3 / 4) * a^2 Thus, the area of the hexagon is: A_hexagon = 6 * A_triangle = 6 * (√3 / 4) * a^2

Step-by-Step Solution:
Let side a = 6 cm. Compute area of one equilateral triangle: A_triangle = (√3 / 4) * a^2. So A_triangle = (√3 / 4) * 6^2 = (√3 / 4) * 36. Simplify: 36 / 4 = 9, so A_triangle = 9√3 sq cm. Area of hexagon = 6 * A_triangle = 6 * 9√3. Therefore, A_hexagon = 54√3 sq cm.

Verification / Alternative check:
There is also a direct formula for a regular hexagon: A_hexagon = (3√3 / 2) * a^2 Substitute a = 6: A_hexagon = (3√3 / 2) * 36 = 3√3 * 18 = 54√3. This matches the result from the equilateral triangle decomposition, confirming the accuracy.

Why Other Options Are Wrong:
27 and 54 with no √3 ignore the surd factor that naturally appears in equilateral triangle area. 27√3 is exactly half the correct result and would come from mistakenly using three triangles instead of six or misapplying the formula.

Common Pitfalls:
A frequent mistake is to forget that the hexagon splits into six equilateral triangles, not four or three. Another common error is miscomputing the area of a single equilateral triangle, especially the factor (√3 / 4). Careful use of the formula avoids these errors.

Final Answer:
The area of the regular hexagon is 54√3 sq cm.