In plane geometry, what is the area, in square centimetres (cm^2), of a regular hexagon in which each side measures 9 cm, expressed in exact surd form?

Introduction / Context:
This question tests knowledge of the area formula for a regular hexagon and how to apply it correctly when the side length is given. Regular polygons appear very often in aptitude and geometry exams, and the regular hexagon is especially important because it can be broken into six congruent equilateral triangles. Understanding this structure allows you to compute the area in an efficient and systematic way.

Given Data / Assumptions:

• The polygon is a regular hexagon, so all six sides and all interior angles are equal.
• The length of each side is 9 cm.
• We are asked to find the total area in square centimetres, written in exact surd form involving sqrt(3).
• No rounding or approximation is required; we keep exact values.

Concept / Approach:
A regular hexagon with side length a can be divided into 6 congruent equilateral triangles, each with side a. The area of a single equilateral triangle with side a is (sqrt(3) / 4) * a^2. Therefore, the area of the regular hexagon is 6 times that triangle area. Another equivalent formula is: Area of regular hexagon = (3 * sqrt(3) / 2) * a^2. Both ways give the same result when applied correctly.

Step-by-Step Solution:
Let the side length of the regular hexagon be a = 9 cm.Formula for the area of a regular hexagon: Area = (3 * sqrt(3) / 2) * a^2.Compute a^2: a^2 = 9^2 = 81.Substitute into the formula: Area = (3 * sqrt(3) / 2) * 81.Multiply the coefficient: 3 * 81 = 243, so Area = 243 * sqrt(3) / 2 square centimetres.Thus the exact area is 243√3/2 cm^2.

Verification / Alternative check:
Alternative method: split the hexagon into 6 equilateral triangles of side 9 cm. Area of one equilateral triangle = (sqrt(3) / 4) * 9^2 = (sqrt(3) / 4) * 81 = 81 * sqrt(3) / 4. Multiply by 6: Area = 6 * (81 * sqrt(3) / 4) = (486 * sqrt(3)) / 4 = 243 * sqrt(3) / 2 after simplifying. This matches the earlier formula, confirming the correctness of the result.

Why Other Options Are Wrong:

• 50√3 and 100√3 are much smaller than the correct area for a relatively large side length of 9 cm.
• 300√3 and 200√3 overshoot the correct value and do not match any standard formula for a regular hexagon.
• Only 243√3/2 follows directly from the correct area formula.

Common Pitfalls:

• Forgetting that the hexagon splits into six equilateral triangles and using the triangle formula incorrectly.
• Using a wrong formula such as that for a circle or hexagon perimeter instead of area.
• Squaring the side incorrectly or dropping the factor 1/2 when simplifying.

Final Answer:
243√3/2