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2015d.00777 Skurnick, Ronald Baderian, Armen Javadi, Mohammad Relating the interior of certain geometric figures to their boundary via differentiation. Math. Comput. Educ. 49, No. 1, 33-46 (2015). 2015 MATYC Journal, Old Bethpage, NY EN I40 I60 G30 G90 differential calculus derivatives plane geometry solid geometry \$n\$-dimensional Euclidean space convex regions area perimeter volume surface area regular polygons Platonic solids inradius circumradius polygons polyhedra inscribed circles circumscribed circles inscribed spheres circumscribed spheres hyperspheres From the text: When you take the derivative of the area of a circle with respect to its radius \$r\$, you get the circumference of the circle. Similarly, when you take the derivative of the volume of a sphere with respect to its radius \$r\$, you obtain the surface area of the sphere. As you may suspect, or already know, these two results are not coincidental, and can be explained using elementary geometric arguments. Perhaps the following presentation will be of interest to a mathematics club or an, individual student who has completed a year of calculus and has studied Euclidean plane and solid geometry, and trigonometry. In this article, we shall see that the above results are just two examples of a more general phenomenon.