What is the area (in square centimetres) of a regular hexagon whose each side measures 14 cm?

Introduction / Context:
This problem deals with the area of a regular hexagon, which is a six sided polygon with all sides and angles equal. For a regular hexagon, there is a standard formula for area in terms of the side length. Alternatively, the hexagon can be divided into six equilateral triangles, and the area can be computed by summing the areas of these triangles. This question tests knowledge of these geometric facts and the ability to apply them to a side length of 14 cm.

Given Data / Assumptions:
- The hexagon is regular (all sides and angles equal).- Each side length a = 14 cm.- We use the standard area formula for a regular hexagon.- Alternatively, we may use the area of equilateral triangles to derive the same result.

Concept / Approach:
A regular hexagon can be divided into 6 congruent equilateral triangles, each having side length equal to the hexagon side a. The area of an equilateral triangle with side a is (√3 / 4) * a^2. Therefore, the area of a regular hexagon is 6 times this: Area_hexagon = 6 * (√3 / 4) * a^2 = (3√3 / 2) * a^2. We substitute a = 14 into this formula to find the required area. This approach makes use of both polygon decomposition and formula application.

Step-by-Step Solution:
Step 1: Let side length a = 14 cm.Step 2: Use the area formula for a regular hexagon: Area = (3√3 / 2) * a^2.Step 3: Compute a^2 = 14^2 = 196.Step 4: Substitute into the formula: Area = (3√3 / 2) * 196.Step 5: Multiply 3 and 196: 3 * 196 = 588.Step 6: So Area = (588 / 2) * √3.Step 7: Simplify 588 / 2 = 294.Step 8: Therefore, Area = 294√3 square centimetres.

Verification / Alternative check:
As a check, we can explicitly consider the smaller equilateral triangles. Each equilateral triangle has area (√3 / 4) * a^2. With a = 14, a^2 = 196, so area of one triangle = (√3 / 4) * 196 = 49√3. There are 6 such triangles in the hexagon, so Area_hexagon = 6 * 49√3 = 294√3 square centimetres. This matches the result from the direct formula and confirms the correctness of the calculation.

Why Other Options Are Wrong:
- 147√3: This is exactly half of the correct area and would correspond to only three equilateral triangles instead of six.- 441√3: This is one and a half times the correct area and does not correspond to any standard decomposition.- 196√3: This would correspond to only four equilateral triangles of side 14 instead of six.- 168√3: This gives a smaller incorrect area, not matching any integer multiple of 49√3 except 49 * 3.428, which lacks geometric meaning here.

Common Pitfalls:
Some learners mistakenly use the area formula for a regular polygon incorrectly or forget that a regular hexagon can be divided into six equilateral triangles. Others may miscompute a^2 or mishandle the fraction 3 / 2 in the formula, leading to numerical errors. Remembering that a regular hexagon has 6 sides and decomposes into 6 congruent equilateral triangles is a helpful visualization that reinforces the area formula.

Final Answer:
The area of the regular hexagon is 294√3 square centimetres.