直观感受：扇形面积

Imagine it's Thanksgiving and you're eating pie. Now, you're not eating the whole pie, right? You're only eating a piece of it. Now assume the cuts of the pie all go through the center. So the piece you're eating is a fraction $\frac{a}{b}$ of the pie from the center. Now imagine the pie is a circle with radius $r$. The area of the circle is $\pi r^2$. The part you are eating has area $\frac{a\pi r^2}{b}$. Get it?

A sector of a circle is an area on the circle that is within two radii and the arc between them, like so:

(Here, an arc is the area enclosed within the red lines and the arc they share.)

The most common way to calculate the area of a sector is to find its proportion to the area of the entire circle. This can be done by finding the ratio of the inner angle of the sector to $360^\circ$.

For example, the sector above appears to have an inner angle of $90^\circ$. $90^\circ$ is $\frac{1}{4}^{th}$ of $360^\circ$, so the area of the sector is $\frac{1}{4}^{th}$ the area of the circle.

Here's a great video reviewing the definition of a sector and showing how to find its area!