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 2πr, and the area of a parallelogram is determined using a = bh, which represents the approximate area of the parallelogram - like figure? \bigcirc a = (2πr)(r) \bigcirc a = (2πr)(2r) \bigcirc a = \frac{1}{2}(2πr)(r) \bigcirc a = \frac{1}{2}(2πr)(r^2)

Explanation:

Step1: Analyze the base and height

When a circle is divided into sectors and rearranged into a parallelogram - like shape, the base of the parallelogram - like figure is half of the circumference of the circle. The circumference of the circle is \( C = 2\pi r \), so the base \( b=\frac{1}{2}(2\pi r) \). The height \( h \) of the parallelogram - like figure is equal to the radius \( r \) of the circle.

Step2: Apply the area formula of parallelogram

The area formula of a parallelogram is \( A = bh \). Substituting \( b=\frac{1}{2}(2\pi r) \) and \( h = r \) into the formula, we get \( A=\frac{1}{2}(2\pi r)(r) \).

Answer:

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