ABCDEF is a regular hexagon with each side equal to 12 cm. What is the area, in square centimetres, of triangle ECD?

Introduction / Context:
This problem involves a regular hexagon and asks for the area of a triangle formed by three of its vertices. Regular polygons have strong symmetry, and in the case of a regular hexagon, many internal triangles are equilateral or easily related to equilateral triangles. By using these symmetries and basic area formulas, we can find the area of triangle ECD without needing heavy coordinate calculations.

Given Data / Assumptions:

• ABCDEF is a regular hexagon.
• Each side length a = 12 cm.
• Vertices are in order around the hexagon.
• We need the area of triangle ECD.
• In a regular hexagon, all internal angles are 120 degrees and all sides are equal.

Concept / Approach:
It helps to visualise the regular hexagon with centre O and vertices A, B, C, D, E, F equally spaced on a circle. Consecutive vertices form equilateral triangles with the centre. For triangle ECD, vertices C and D are adjacent, while E is adjacent to D on the other side. Thus triangle CDE has two sides that are hexagon sides and an included interior angle of 120 degrees at D. We can use the formula for the area of a triangle with two sides and the included angle: (1/2) * a * b * sin(angle).

Step-by-Step Solution:
In regular hexagon ABCDEF, side length a = 12 cm. Consider triangle CDE with vertices C, D and E on the hexagon. Sides CD and DE are both sides of the hexagon, so CD = DE = 12 cm. The interior angle at D in the hexagon is 120 degrees, and this is the angle between CD and DE in triangle CDE. Area of a triangle given two sides and included angle: Area = (1/2) * a * b * sin(theta). So Area(CDE) = (1/2) * 12 * 12 * sin(120 degrees). Compute sin(120 degrees) = sin(60 degrees) = square root of 3 / 2. Area = (1/2) * 144 * (square root of 3 / 2) = 72 * (square root of 3 / 2) = 36 * square root of 3.

Verification / Alternative check:
Another way is to recognise that a regular hexagon can be divided into six congruent equilateral triangles of side 12 cm by drawing all radii from the centre. The area of each equilateral triangle is (square root of 3 / 4) * a^2 = (square root of 3 / 4) * 144 = 36 * square root of 3. In a regular hexagon, some triangles formed by non adjacent vertices coincide with these fundamental triangles. Triangle CDE is such a fundamental equilateral based triangle, so its area equals 36 * square root of 3 sq cm, which matches the previous method.

Why Other Options Are Wrong:
18√3 and 24√3: These are smaller than the actual area and would arise from halving the result or using a shorter side length. 42√3 and 30√3: These do not correspond to any simple multiple of the equilateral triangle area with side 12 cm and are inconsistent with regular hexagon geometry.

Common Pitfalls:
Misidentifying which triangles in the hexagon are equilateral and which are not. Using the wrong interior angle at D, such as 60 degrees instead of 120 degrees. Forgetting that sin(120 degrees) equals sin(60 degrees) and not cos(60 degrees).

Final Answer:
The area of triangle ECD is 36√3 square centimetres