Triangle from alternate vertices of a regular hexagon (side 12 cm):\nPQRSTU is a regular hexagon of side 12 cm. Find the exact area (in sq cm) of triangle SQU.

Introduction / Context:
In a regular hexagon, connecting every other vertex forms an equilateral triangle. Determining the side of that triangle in terms of the hexagon’s side gives the area directly via the equilateral area formula.

Given Data / Assumptions:

• Regular hexagon with side a = 12 cm.
• Vertices in order P-Q-R-S-T-U; triangle uses S, Q, U (alternate vertices).

Concept / Approach:
Coordinate placement shows the distance between vertices two apart equals √3 * a, making triangle SQU equilateral with side √3 * a. Area of an equilateral triangle with side t is (√3/4) * t^2.

Step-by-Step Solution:

Verification / Alternative check:
Direct coordinate computation with a unit hexagon scaled by 12 confirms all three sides equal 12√3.

Why Other Options Are Wrong:
Other multiples do not match (√3/4)*(12√3)^2.

Common Pitfalls:
Assuming the triangle side equals 2a (that would be for opposite vertices through the center), not for alternate vertices.

Final Answer:
108√3