The Peculiar Equation

Geometry Level 2

The area of a circle is known to be $A=\pi r^2$ . If we were to take the derivative with respect to $r$ , we obtain the circumference of the circle, namely $C=2\pi r$ .Now, consider any $2D$ shape whose area is given by $A(x)$ .

True or False: $\dfrac{dA}{dx}=\text{perimeter} \\ \text{is always true}$

True False

1 solution

Hamza A
Jan 22, 2019

Consider a square of side $x$ , $\frac{dA}{dx}=2x\neq \text{Perimeter}=4x$

Interestingly, the relationship can be established if an appropriate linear dimension is established. Set $s=\frac{x}{2}$ and we find that that the derivative of the area indeed does equal the perimeter. My conjecture is below:

Conjecture :

For any shape with a well-defined, finite area and perimeter we can parametrize $A(x)$ such that

$\dfrac{dA}{dx}=\text{perimeter}$

I also suspect that the same argument can be made about the relationship of volume and surface area.

@Hummus a A small typo, I think you meant s=2x instead of s=x/2...........

Aaghaz Mahajan - 2 years, 4 months ago

There is no typo

Hamza A - 2 years, 4 months ago

Well, if $s=\frac{x}{2}$ then, acc. to you, the area of the square would be $\frac{x^2}{4}$ whose derivative is $\frac{x}{2}$ which DOES NOT equal the perimeter of the square with side length $s$ i.e. $2x$ .

Whereas, taking $s=2x$ , the area would be $4x^2$ whose derivative is $8x$ which IS equal to the perimeter of the square with side length $2x$

Aaghaz Mahajan - 1 year, 11 months ago

The Peculiar Equation

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