Derivative of Area Equals Perimeter - Coincidence or Rule?

Why is the derivative of the area of a circle equal to its circumference? Why is the derivative of the volume of a sphere equal to its surface area? And why does a similar relationship not hold for a square or a cube? Or does it? In their work in teacher education, these authors have heard at times undesirable responses to these questions: "That's the way it is. Circles and spheres are very special. Squares and cubes have corners." Or, "It is a simple coincidence with circles. This relationship does not hold for any other shapes." This article explores and explains the familiar relationship of the area of a circle and its circumference and of the volume of a sphere and its surface area. It then extends this relationship to other two- and three-dimensional figures--squares and regular polygons, cubes and regular polyhedra.