If each side of the octagon is a, find the area of the octagon.

Options

A. `2(sqrt2-1)a^2`
B. `4(1+sqrt2)a^2`
C. `2(2+sqrt2)a^2`
D. `2(1+sqrt2)a^2`

Answer

To find the area of a regular octagon where each side is of length \( a \), we can use the formula for the area of a regular octagon:

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

### Step-by-Step Solution:

1. **Identify the formula for the area of a regular octagon**:
The area \( A \) of a regular octagon can be expressed as:
\[ A = 2(1 + \sqrt{2})a^2 \]

2. **Substitute the value of \( a \)**:
Since each side of the octagon is given as \( a \), we can directly substitute \( a \) into the formula:
\[ A = 2(1 + \sqrt{2})a^2 \]

3. **Simplify the expression**:
The expression \( 1 + \sqrt{2} \) is a constant, so we can keep it as is. The area in terms of \( a \) is:
\[ A = 2(1 + \sqrt{2})a^2 \]

4. **Final expression**:
Thus, the area of the octagon in terms of side length \( a \) is:
\[ A = 2(1 + \sqrt{2})a^2 \]