If area of regular octagon is `18 ( sqrt(2)+ 1 )` . Find this side

Answer

To find the side length of a regular octagon given its area, we can use the formula for the area of a regular octagon:

\[ \text{Area} = 2a^2(\sqrt{2} + 1) \]

where \( a \) is the length of a side of the octagon.

Given that the area is \( 18(\sqrt{2} + 1) \), we can set up the equation:

\[ 2a^2(\sqrt{2} + 1) = 18(\sqrt{2} + 1) \]

### Step 1: Simplify the equation
Since both sides of the equation have the term \( \sqrt{2} + 1 \), we can divide both sides by \( \sqrt{2} + 1 \) (assuming \( \sqrt{2} + 1 \neq 0 \)):

\[ 2a^2 = 18 \]

### Step 2: Solve for \( a^2 \)
Now, we can solve for \( a^2 \) by dividing both sides by 2:

\[ a^2 = \frac{18}{2} = 9 \]

### Step 3: Solve for \( a \)
Now we take the square root of both sides to find \( a \):

\[ a = \sqrt{9} = 3 \]

Thus, the side length of the regular octagon is:

\[ \boxed{3} \]