Question

the diagram shows one way to develop the formula for the area of a circle. pieces of a circle with radius r are rearranged to create a shape that resembles a parallelogram. since the circumference of the circle can be represented by ( 2pi r ), and the area of a parallelogram is determined using ( a = bh ), which represents the approximate area of the parallelogram-like figure? (\bigcirc) ( a = (2pi r)(r) ) (\bigcirc) ( a = (2pi r)(2r) ) (\bigcirc) ( a = \frac{1}{2}(2pi r)(r) ) (\bigcirc) ( a = \frac{1}{2}(2pi r)(r^2) )

Explanation:

Step1: Analyze the base and height of the parallelogram

When a circle is divided into sectors and rearranged into a parallelogram - like shape, the base of the parallelogram is approximately half of the circumference of the circle? Wait, no. Wait, actually, when we rearrange the sectors of the circle, the length of the base of the parallelogram - like figure is half of the circumference of the circle? Wait, no, let's think again. The circumference of the circle is \( C = 2\pi r \). When we cut the circle into many small sectors and rearrange them, the base of the parallelogram - like figure is \( \frac{1}{2}\times2\pi r=\pi r \)? Wait, no, maybe I made a mistake. Wait, the problem says "the circumference of the circle can be represented by \( 2\pi r \)", and the area of a parallelogram is \( A = bh \). Let's look at the height of the parallelogram - like figure. The height of this figure is equal to the radius \( r \) of the original circle? Wait, no, when we rearrange the sectors, if we have a shape that resembles a parallelogram, the base of the parallelogram is half of the circumference? Wait, no, let's check the options.

Wait, the formula for the area of a parallelogram is \( A=bh \). Let's find the base and height of the parallelogram - like figure formed from the circle. The circle has circumference \( C = 2\pi r \). When we divide the circle into sectors and rearrange them, the length of the base of the parallelogram - like figure is \( \frac{1}{2}\times2\pi r=\pi r \)? No, wait, maybe the base is \( \frac{1}{2}(2\pi r) \) and the height is \( r \). Wait, let's check the options.

Option 1: \( A=(2\pi r)(r) \)
Option 2: \( A=(2\pi r)(2r) \)
Option 3: \( A = \frac{1}{2}(2\pi r)(r) \)
Option 4: \( A=\frac{1}{2}(2\pi r)(r^{2}) \)

Wait, when we rearrange the sectors of the circle to form a parallelogram - like shape, the base of the parallelogram is half of the circumference of the circle. The circumference is \( 2\pi r \), so half of it is \( \frac{1}{2}(2\pi r) \). The height of the parallelogram is equal to the radius \( r \) of the circle. Then, using the area formula for a parallelogram \( A = bh \), where \( b=\frac{1}{2}(2\pi r) \) and \( h = r \), so \( A=\frac{1}{2}(2\pi r)(r) \).

Step2: Evaluate each option

• Option 1: \( A=(2\pi r)(r) \) assumes the base is \( 2\pi r \), but the base of the parallelogram - like figure is half of the circumference, so this is incorrect.
• Option 2: \( A=(2\pi r)(2r) \) has an incorrect base and height, so this is wrong.
• Option 3: \( A=\frac{1}{2}(2\pi r)(r) \) uses the correct base (half of the circumference) and height (radius), so this is correct.
• Option 4: \( A=\frac{1}{2}(2\pi r)(r^{2}) \) has an incorrect height (it uses \( r^{2} \) instead of \( r \)), so this is wrong.

Answer:

\( A = \frac{1}{2}(2\pi r)(r) \) (corresponding to the third option: \( A=\frac{1}{2}(2\pi r)(r) \))