To derive the area of a circle using the arrangement of the sectors as described, we can follow the reasoning provided.
- The circumference of the circle \( C \) is given by \( C = 2\pi r \).
- When the circle is divided into 8 equal sectors, the arc length of each sector is \( \frac{2\pi r}{8} = \frac{\pi r}{4} \).
- When rearranging the sectors into two rows, we can visualize this as forming a shape that resembles a parallelogram.
- The height of the parallelogram formed is the radius \( r \), and the base length (which comes from the arc lengths of the sectors) can be approximated as half the circumference, which is \( \pi r \) (given that each row contains 4 pieces).
Now, to calculate the area \( A \) of the parallelogram formed, we use the formula for the area of a parallelogram, which is:
\[ A = \text{base} \times \text{height} \]
Here, the base \( = \pi r \) and the height \( = r \).
So, we have:
\[ A = \pi r \cdot r = \pi r^2 \]
Thus, the area \( A \) of the circle is:
\[ A = \pi r^2 \]
Given your options, the correct representation for the area of the circle is:
A. \( A = \pi r \cdot r^2 \)
(Note: Though the wording seems slightly off in this option's presentation regarding \( r^2 \), this is actually consistent with \( \pi r^2 \) since there should be only one \( r \) considering standard form. Usually, it should be written as \( A = \pi r^2 \). This indicates a potential typographical or phrasing issue in the options provided.)