Question

use the diagram to informally derive the formula for area of a circle.
if we cut a circle into sectors and arrange them in a line with alternating orientations, the shape looks like a parallelogram. as the number of sectors increases, the height of this parallelogram
will approach
and the length will approach
therefore, the area approaches

Explanation:

Step1: Recall the formula for the circumference of a circle

The circumference of a circle is \(C = 2\pi r\). When we cut the circle into sectors and arrange them as a parallelogram - like shape, the base of the parallelogram - like shape (as the number of sectors \(n\to\infty\)) approaches half of the circumference of the circle. So, the base \(b=\pi r\) (since \(b=\frac{2\pi r}{2}\)).

Step2: Recall the height of the parallelogram - like shape

The height \(h\) of the parallelogram - like shape approaches the radius \(r\) of the circle as the number of sectors increases.

Step3: Use the formula for the area of a parallelogram

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

Answer:

\(\pi r^{2}\)