Derivative of the area is the circumference -- generalization

In summary, the derivative of the area of a circle is equal to its circumference, and this concept can be extended to all regular polygons. By changing the measurement of the polygon used in the formulas, we can generalize this concept. For example, instead of using the side length, we can use the radius of the square as the independent variable. This can also be seen in the relationship between volume and surface area for a 3-ball, and can be further explored for a 4-ball.
  • #1
1MileCrash
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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.

 
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  • #2
1MileCrash said:
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
 
  • #3
Buzz Bloom said:
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.
 
  • #4
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?
 
  • #5
"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.
 
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Related to Derivative of the area is the circumference -- generalization

1. What is the derivative of the area formula for a circle?

The derivative of the area formula for a circle is equal to the circumference of the circle.

2. How is it possible for the derivative of the area to be the circumference?

This happens because the area and circumference of a circle are directly related. As the radius of a circle increases, so does its circumference, which in turn increases the area of the circle. Therefore, the derivative of the area with respect to the radius is equal to the circumference.

3. Can this generalization be applied to other shapes besides circles?

Yes, this generalization can be applied to any shape that has a direct relationship between its area and perimeter. For example, the derivative of the area formula for a square is equal to its perimeter.

4. How can this generalization be useful in real-world applications?

Knowing the derivative of the area formula can be useful in calculating the rate of change of a shape's area with respect to its perimeter. This can be helpful in fields such as geometry, engineering, and physics.

5. Is it possible for the derivative of the area to be different from the circumference in certain cases?

Yes, in some cases, the derivative of the area may not be equal to the circumference. This typically happens in irregular or non-geometric shapes, where there is no direct relationship between the area and perimeter.

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