Explain please Area of circle

[The Blade Dancer]

Area of circle and stuff I am sooo confused about. Not really in any of the courses I've taken, but I've seen it and I'm confused.

[nastya]

Hi @The-Darkin-Blade!
Nice to hear from you again!
The area of circle is a veeery big topic for a small forum post. There are lots of articles about the formula of a circle's area and other things related to this topic! Hundreds! Even thousands!
Your confusion is normal
To learn more about it, you can start from here, or find more advanced information here.

You can also imagine that a circle is a regular polygon with lots and lots of vertices (actually, infinitely many vertices). To find the area of a regular polygon, you cut it into isosceles triangles (see picture) and then sum up their areas. So the area of a polygon is equal to $n\times(\frac{1}{2}ah)=\frac{h}{2}\times(na),$ where $n$ - number of vertices, $a$ - side length and $h$ - distance from the center of polygon to its side. The term $na$ is the perimeter of the polygon. As the polygon becomes more and more like a circle, this value approaches the value of the circle's circumference, which is $2\pi r,$ and value of $h$ approaches the circle radius. So, substituting $2\pi r$ instead of $na,$ we get: $\text{area of circle}=\frac{r}{2}\times(2\pi r)=\pi r^2.$

Studying this in more detail and discovering lots of new and very interesting things will be easier as you get older and study higher-level topics like functions, derivatives, integrals, and many other things.