One of my favorite examples is to explain why the derivative of the area of a circle is the circumference. And the derivative of the volume of a sphere is the surface area. If you try the same for the square and cube, it may not work at first, but try to not use the length of the side as the parametere, but half the length of the side.

You can argue in many ways. My favorite is to cut the ring and bend it out to get something that looks like rectangle with width equal to the circumference, and height equal to \delta r.

One of my favorite examples is to explain why the derivative of the area of a circle is the circumference. And the derivative of the volume of a sphere is the surface area. If you try the same for the square and cube, it may not work at first, but try to not use the length of the side as the parameter, but half the length of the side.

You can argue in many ways. My favorite is to cut the ring and bend it out to get something that looks like rectangle with width equal to the circumference, and height equal to $\Delta r$.