Question

professor smart is teaching her students how the formula for the area of a circle was derived. she cuts a circle into equal pieces and rearranges them below.
a. what general 2d figure do you see when the pieces are rearranged? what is the formula for the area of that shape?
b. based on the diagram above, how is the circumference of the circle related to the length of the parallelogram?

Explanation:

a.

Step1: Identify the general 2D figure

When the pieces of the circle are rearranged, the general 2D figure formed is a parallelogram.

b.

Step1: Relate the circumference and the parallelogram's side

The circumference of the circle \(C = 2\pi r\). In the parallelogram formed from the circle segments, the length of the parallelogram (the base) is related to half of the circumference of the circle. So, if the circumference is \(C\), then the base \(b\) of the parallelogram is \(\frac{C}{2}=\pi r\). The height \(h\) of the parallelogram is equal to the radius \(r\) of the circle.

Step2: Recall the area formula of a parallelogram

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

Answer:

a. A parallelogram.
b. The circumference of the circle \(C = 2\pi r\). The base of the parallelogram \(b=\frac{C}{2}=\pi r\), and the height \(h = r\). Using the parallelogram area formula \(A = b\times h\), substituting \(b\) and \(h\) gives \(A=\pi r^{2}\), showing the relationship.