Derivative of the area is the circumference -- generalization

[1MileCrash]

I thought you guys might appreciate this. A lot of people notice that the derivative of area of a circle is the circle's circumference. This can be generalized to all regular polygons in a nice way.

 

[Buzz Bloom]

This can be generalized to all regular polygons in a nice way.

Hi 1MileCrash:

I don't get this. What is the nice way?
The area of a square with side length x is x².
The derivative is 2x. The circumference = boundary length is 4x.

Regards,
Buzz

 

[1MileCrash]

Hi 1MileCrash:

I don't get this. What is the nice way?
The area of a square with side length x is x².
The derivative is 2x. The circumference = boundary length is 4x.

Regards,
Buzz

You've made a choice to express the formulas in terms of the square's side length, but there's nothing wrong with expressing the formulas in terms of some other measurement of the square.

The generalization is found by changing the measurement of the polygon that we express the formulas in terms of.

 

[mathwonk]

nice comment.
try using the "radius" of the square, r = x/2 as independent variable. and for a 3-ball, consider the relationship between volume and surface area. what about a 4-ball?

 

[Varkman]

"the derivative of the area formula for a square is equal to its perimeter."

This make little sense to me. When the side equals x, the area is x^2, and the perimeter is 4x.

The derivative of x^2 is 2x, not 4x.

EDIT: I see there were other responses like mine. The text should be changed from the generalization so as not to distract the reader.