Neither it's the longest one nor it's the shortest one of the lengths of the diagonal of an octagon you know is d and each side of the same octagon is s, find the area of the octagon in terms of the side and the known diagonal of the regular octagon.

Options

A. `3d^2-sqrt3s^2`
B. `d^2-s^2`
C. 2ds
D. BOTH (B) AND (C)

Answer

To solve the problem of finding the area of a regular octagon in terms of its side length \( s \) and a known diagonal length \( d \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Properties of a Regular Octagon**:
A regular octagon has 8 equal sides and equal angles. The area of a regular octagon can be expressed in terms of its side length \( s \).

2. **Formula for the Area of a Regular Octagon**:
The area \( A \) of a regular octagon can be calculated using the formula:
\[ A = 2(1 + \sqrt{2})s^2 \]

3. **Formula for the Diagonal of a Regular Octagon**:
The length of a diagonal \( d \) of a regular octagon can be expressed in terms of the side length \( s \). The formula for one of the diagonals is:
\[ d = s(1 + \sqrt{2}) \]

4. **Express \( 1 + \sqrt{2} \) in terms of \( d \) and \( s \)**:
Rearranging the diagonal formula gives:
\[ 1 + \sqrt{2} = \frac{d}{s} \]

5. **Substituting into the Area Formula**:
Now substitute \( \frac{d}{s} \) into the area formula:
\[ A = 2\left(\frac{d}{s}\right)s^2 \]

6. **Simplifying the Area Expression**:
Simplifying this expression gives:
\[ A = 2ds \]

7. **Conclusion**:
Therefore, the area of the octagon in terms of the side length \( s \) and the diagonal length \( d \) is:
\[ A = 2ds \]

### Final Answer:
The area of the octagon is \( 2ds \).

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