Find the area of a regular octagon with each side 'a' cm :

Options

A. `2a^(2)(1+ sqrt(2))`
B. `sqrt(2)a(1+ pi)`
C. `a^(2) ( sqrt(2) +2)`
D. none of these

Answer

To find the area of a regular octagon with each side measuring 'a' cm, we can use the formula for the area of a regular octagon:

**Step 1: Write down the formula for the area of a regular octagon.**
The area \( A \) of a regular octagon can be calculated using the formula:
\[ A = 2(1 + \sqrt{2})a^2 \]

**Step 2: Substitute the value of 'a'.**
In this case, we are given that each side \( a \) is equal to 8 cm. Therefore, we substitute \( a = 8 \) into the formula:
\[ A = 2(1 + \sqrt{2})(8^2) \]

**Step 3: Calculate \( 8^2 \).**
Calculating \( 8^2 \):
\[ 8^2 = 64 \]

**Step 4: Substitute \( 64 \) back into the area formula.**
Now substitute \( 64 \) into the area formula:
\[ A = 2(1 + \sqrt{2})(64) \]

**Step 5: Simplify the expression.**
Now, we can simplify:
\[ A = 128(1 + \sqrt{2}) \]

**Step 6: Calculate \( 1 + \sqrt{2} \).**
The value of \( \sqrt{2} \) is approximately \( 1.414 \), so:
\[ 1 + \sqrt{2} \approx 1 + 1.414 = 2.414 \]

**Step 7: Multiply \( 128 \) by \( 2.414 \).**
Now, we multiply:
\[ A \approx 128 \times 2.414 \approx 309.312 \]

**Step 8: Finalize the area.**
Thus, the area of the regular octagon with each side measuring 8 cm is approximately:
\[ A \approx 309.31 \, \text{cm}^2 \]