What is the area (in square centimetres) of a regular hexagon whose side length is 6 cm?

Introduction / Context:
This question involves finding the area of a regular hexagon, a six sided polygon with all sides and interior angles equal. Regular hexagons appear in many geometric tiling and design problems. A useful fact is that a regular hexagon can be decomposed into six congruent equilateral triangles, which allows us to use the equilateral triangle area formula to compute the total area of the hexagon efficiently.

Given Data / Assumptions:

• The polygon is a regular hexagon.
• Each side of the hexagon has length 6 cm.
• We must compute the area in square centimetres.
• The hexagon can be divided into six congruent equilateral triangles, each with side 6 cm.
• Standard equilateral triangle area formulas apply.

Concept / Approach:
A regular hexagon of side a can be divided into six equilateral triangles, each with side a. The area of one equilateral triangle with side a is A_triangle = (√3 / 4) * a^2. The total area of the hexagon is then A_hexagon = 6 * A_triangle. Substituting a = 6 into this formula gives us the exact area in terms of √3. There is also a compact formula A_hexagon = (3√3 / 2) * a^2, which is simply the simplified version of 6 * (√3 / 4) * a^2.

Step-by-Step Solution:
Given side length a = 6 cm.Area of one equilateral triangle: A_triangle = (√3 / 4) * a^2.Compute a^2 = 6^2 = 36.So A_triangle = (√3 / 4) * 36 = 9√3 sq cm.A regular hexagon consists of 6 such equilateral triangles, so A_hexagon = 6 * 9√3 = 54√3 sq cm.

Verification / Alternative check:
Using the compact formula for a regular hexagon, A_hexagon = (3√3 / 2) * a^2. Substituting a = 6 gives A_hexagon = (3√3 / 2) * 36. Compute 36 / 2 = 18, so A_hexagon = 3√3 * 18 = 54√3 sq cm, which matches the decomposition approach. This confirms that 54√3 is the correct area.

Why Other Options Are Wrong:
Option 27√3 is exactly half the correct value and corresponds to using only 3 equilateral triangles instead of 6. Option 54 without √3 ignores the radical part and underestimates the area. Option 27 is again too small and also lacks √3. Option 81√3 is larger than the correct area and might result from incorrectly multiplying by 9 instead of 6. Only 54√3 correctly follows from the known formulas for regular hexagons.

Common Pitfalls:
Some students incorrectly think that a regular hexagon can be treated as a rectangle or that its area is simply side squared multiplied by 6. Others forget that the internal triangles are equilateral and therefore use wrong area formulas. Miscounting the number of triangles or miscomputing 6 * 9 leads to errors as well. Remembering the simple decomposition into six equilateral triangles makes this type of problem straightforward.

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