Question

use the diagram to informally derive the formula for area of a circle.
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 area of a parallelogram

The area of a parallelogram is \(A = base\times height\).

Step2: Identify the base and height of the parallelogram formed by the circle sectors

• When we rearrange the sectors of a circle to form a parallelogram - like shape, the base of the parallelogram approaches \(\frac{1}{2}\times(2\pi r)=\pi r\) (half of the circumference of the circle, since the circumference \(C = 2\pi r\)).
• The height of the parallelogram approaches the radius \(r\) of the circle.

Step3: Calculate the area of the parallelogram (and thus the area of the circle)

Substitute the base \(b=\pi r\) and height \(h = r\) into the area formula \(A=b\times h\). So \(A=\pi r\times r=\pi r^{2}\).

Answer:

\(\pi r^{2}\)