Summary As is shown by the large literature on this subject, we are not the first to be surprised by the fact that, for the circle, the derivative of the area is equal to its perimeter. Our point of view to better understand this seemingly miraculous relationship, comes from differential geometry, leading us to necessary thoughts on what a derivative is and the major role played by changes of coordinates. Moreover, this use of differential geometry seems unavoidable when it comes to studying the case of figures depending on several parameters. Our approach to these questions will take place in the framework of manifolds of figures and will use the notions of directional derivatives and deformations of figures.
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