A regular octagon ABCDEFGH has sides of length 6 cm each. Then the area, in square cm, of the square ACEG is:

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1. **Interior angle of the octagon:**  
   - A regular octagon has an interior angle calculated as:  
     $$ 
     	ext{Interior angle} = \frac{n-2}{n} 	imes 180^\circ = \frac{8-2}{8} 	imes 180^\circ = 135^\circ.
     $$

2. **Using the cosine rule in triangle $$ 	riangle BAC $$:**  
   - The square ACEG is formed by connecting alternate vertices of the octagon. To find its side length $$ a $$, use the cosine rule in $$ 	riangle BAC $$:  
     $$ 
     a^2 = b^2 + c^2 - 2bc\cos C,
     $$  
     where $$ b = c = 6 $$ (side length of the octagon), and $$ C = 135^\circ $$.

3. **Substitute values into the cosine rule:**  
   $$ 
   a^2 = 6^2 + 6^2 - 2(6)(6)\cos 135^\circ.
   $$  
   - Since $$ \cos 135^\circ = -\frac{1}{\sqrt{2}} $$:  
     $$ 
     a^2 = 36 + 36 - 72(-\frac{1}{\sqrt{2}}).
     $$  
   - Simplify:  
     $$ 
     a^2 = 72 + 72(\frac{1}{\sqrt{2}}).
     $$  
   - Rationalize:  
     $$ 
     a^2 = 36(2 + \sqrt{2}).
     $$

4. **Area of square ACEG:**  
   - The area of the square is given by $$ a^2 $$:  
     $$ 
     	ext{Area} = 36(2 + \sqrt{2}).
     $$

Thus, the area of the square ACEG is **$$ 36(2 + \sqrt{2}) $$**.

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