Question

1. the picture shows a circle divided into 8 equal wedges which are rearranged. the radius of the circle is r and its circumference is 2πr. how does the picture help to explain why the area of the circle is πr²?

Explanation:

Step1: Analyze the transformed shape

When the circle is divided into 8 equal wedges and rearranged, the shape resembles a parallelogram. The base of this parallelogram is approximately half of the circumference of the circle ($\frac{2\pi r}{2}=\pi r$), and the height is approximately the radius $r$ of the circle.

Step2: Use the area formula of parallelogram

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

Answer:

The picture helps to explain the area formula of the circle because when the circle is transformed (by dividing into wedges and rearranging), it forms a shape (like a parallelogram) whose area formula ($A = base\times height$) can be related to the circle's properties. The base of the transformed shape is about half - circumference ($\pi r$) and height is $r$, so using the parallelogram area formula $A=\pi r\times r=\pi r^{2}$, which is the area formula of the circle.