Solution: Let original side be $ s $. Original area: $ \frac{\sqrt{3}}{4} s^2 $. New side $ s - 3 $, new area: $ \frac{\sqrt{3}}{4} (s - 3)^2 $. The difference: $ \frac{\sqrt{3}}{4} [s^2 - (s - 3)^2] = 15\sqrt{3} $. Simplify: $ \frac{\sqrt{3}}{4} (6s - 9) = 15\sqrt{3} $. Cancel $ \sqrt{3} $ and solve $ \frac{6s - 9}{4} = 15 $, leading to $ 6s - 9 = 60 $, so $ s = \frac{69}{6} = 11.5 $. Original side length is $ \boxed{11.5} \, \text{cm} $.

["Optimize Your Hexagon Area with a Simple Mathematical Solution", "Twenty-sided shape optimization often hinges on precise geometric adjustments. In this case, consider a regular hexagon with original side length $ s $. Its area is given by $ \frac{\sqrt{3}}{4} s^2 $, a formula rooted in symmetry and efficiency.", "Now imagine reducing each side by 3 cm—new side length becomes $ s - 3 $. The updated area is $ \frac{\sqrt{3}}{4} (s - 3)^2 $. The difference in area reveals valuable insights into how diminishing side length affects overall space.", "The area reduction is calculated as:\n$$\n\frac{\sqrt{3}}{4} \left[ s^2 - (s - 3)^2 \right] = 15\sqrt{3}\n$$", "Divide both sides by $ \sqrt{3} $ to simplify:\n$$\n\frac{1}{4} \left[ s^2 - (s - 3)^2 \right] = 15\n$$", "Expand $ (s - 3)^2 = s^2 - 6s + 9 $, then compute the difference:\n$$\ns^2 - (s^2 - 6s + 9) = 6s - 9\n$$", "Plug this back:\n$$\n\frac{1}{4} (6s - 9) = 15\n$$", "Multiply both sides by 4:\n$$\n6s - 9 = 60\n$$", "Solve for $ s $:\n$$\n6s = 69 \quad \Rightarrow \quad s = \frac{69}{6} = 11.5\n$$", "The original side length of the hexagon is therefore $ \boxed{11.5} , \ ext{cm} $. This precise adjustment illustrates how small changes in side length yield predictable area differences—an essential insight in geometry and design.", "Use this method to refine dimensions for optimal space efficiency in architectural, architectural, or mathematical modeling contexts."]