The area of circles is derived by dividing a circle into an infinite number of wedges formed by radii drawn from the center. When these wedges are rearranged, they form a rectangle whose height is the radius of the circle and whose base length is half of the circumference of the circle. The area of circles are also used in sectors, segments and annuluses.
If we want to find the area bound by a circle, well we're going to have to chop this up into pieces of something that we know. And we know the area of a parallelogram is equal to its base times its corresponding height.
So what I'm going to do is I'm going to imagine that I can cut this circle up into an infinite number of pieces. But since I can't draw an infinite number of pieces, I'm just going to start by drawing in let's say about 16 different pieces here. So if I could cut out all these pieces and rearrange them, what I would be able to do is I could, let's say we just looked at one of these pieces. So you're going to have the little curvature on the outside form the circle and that both of these are going to be a radius. So what I'm going to do is I'm going to set all of these up in opposing order and if I just continue doing this, I would be able to from a parallelogram where each of these is going to be one of the wedges that I cut out.
So I said that my height here is going to be r and I need to find out what is my corresponding base. Well, if I look at this, this outside is part of my circle and this outside right here is also the other part of the circle. So if the whole circumference is equal to 2 pi r, then one of these bases is going to be half of this. And, if I divide this by 2, then half of my circumference is just pi times r. So this distance right here is pi times my radius. So if I know my height is r, and I know my base is pi times r, then I can say the area is just r times r which is r squared times pi.
So if I could cut this up into a infinite testable amount of small wedges, I could create a rectangle where I know that my height is r and one of my bases is pi times r and these areas would be equal. So I can say that the area of my circle, where I know the radius is pi times the radius squared.