- #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.
Hi 1MileCrash:1MileCrash said:This can be generalized to all regular polygons in a nice way.
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 x^{2}.
The derivative is 2x. The circumference = boundary length is 4x.
Regards,
Buzz
The derivative of the area formula for a circle is equal to the circumference of the circle.
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.
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.
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.
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.