Question

The figure shows how the formula for the circumference of a circle may be used to derive the formula for the area of a circle.

A circle labeled C equal to 2-pi-r divided into eight equal sectors. The pieces of the circle were separated from one another and arranged in two rows, with four pieces each labelled 2-pi-r over 2. Pieces are brought together into a parallelogram having a height of r.

Based on the figure, which of the following represents the area of the circle?

A.
A
=
π
r
⋅
r
2

B.
A
=
π
r
⋅
r

C.
A
=
2
π
r
⋅
r
2

D.
A
=
2
π
r
⋅
r

Answer 1

To derive the area of a circle using the arrangement of the sectors as described, we can follow the reasoning provided.

1. The circumference of the circle \( C \) is given by \( C = 2\pi r \).
2. When the circle is divided into 8 equal sectors, the arc length of each sector is \( \frac{2\pi r}{8} = \frac{\pi r}{4} \).
3. When rearranging the sectors into two rows, we can visualize this as forming a shape that resembles a parallelogram.
4. 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.)