Area of Rectangle and circle

Question

Just a basic question :

The area of a rectangle is length×width because I'm filling the length with width. This makes sense to me but if I do the same approach for a circle then the area becomes $2\pi$$r^\text{2}$. What am I missing here ?

Answer 1

This doesn't quite work because the circumference is not the same for smaller radii. What you really want to do is to "sum" over all different circumferences $2\pi R$ for $R \leq r$. This is done properly via the integral, i.e. take $2\pi R$ and integrate it over the interval $[0,r]$. Then you get the right result.

Further reading: https://en.wikipedia.org/wiki/Cavalieri%27s_principle

Answer 2

This image might help to understand the circle's area compared with a rectangle: