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? a=(2πr)(r) a=(2πr)(2r) a=\frac{1}{2}(2πr)(r) a=\frac{1}{2}(2πr)(r^{2})

Explanation:

Step1: Find the base and height of the parallelogram - like figure

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

Step2: Use the formula for the area of a parallelogram \(A = bh\)

Substitute \(b=\frac{1}{2}(2\pi r)\) and \(h = r\) into the formula \(A=bh\).

Answer:

\(A=\frac{1}{2}(2\pi r)(r)\)