Regular hexagon ABCDEF (side 12 cm): Find the area (in cm²) of triangle ECD formed by vertices E, C, and D.

Introduction / Context:A regular hexagon of side s can be embedded in a coordinate plane so that exact lengths and heights are easily read. Choosing standard coordinates shows that triangle CDE has a simple base–height structure tied directly to s.

Given Data / Assumptions:

• Regular hexagon ABCDEF with side s = 12 cm.
• Standard placement: A(s,0), B(s/2, √3 s/2), C(−s/2, √3 s/2), D(−s,0), E(−s/2, −√3 s/2), F(s/2, −√3 s/2).

Concept / Approach:With these coordinates, CE is vertical with length √3 s, and D lies horizontally s/2 to the left of the CE line. Thus the altitude from D to CE equals s/2, and the base CE equals √3 s. Area = (1/2)*base*height = (1/2)*(√3 s)*(s/2) = (√3/4) s².

Step-by-Step Solution:

Verification / Alternative check:Since each central triangle of the hexagon has area (√3/4)s², triangle CDE equals exactly one such central triangle (36√3 for s = 12).

Why Other Options Are Wrong:18√3 and 24√3 are too small; 42√3 is too large.

Common Pitfalls:Using side^2 directly without identifying the correct base–height pair; mislabelling vertices.

Final Answer:36√3