The amount of area between the square and circle is given by the difference of the two individual areas, the larger and smaller:

It then holds that the rate of change of this difference in area can be found by taking the time derivative of each side of the equation:

We are told that the difference in area is not changing, which means that . Therefore:

The area of a rectangle is given in terms of its length and width by the formula:

We are asked to find the rate of change of the rectangle when it is a square, i.e at the time that , so we must find the unknown value of  and  at this moment. The width and length at any time can be found in terms of their starting values and rates of change:

When they're equal:

And at this time .

Now, going back to our original area equation

We can take the derivative of each side with respect to time to find the rate of change:

To find , we must first find the derivative and then plug in  for .

To evaluate this derivative, we need the following formulae:

Then plug in  for  into :

The area of a rectangle is given by the function:

For the definitions of the sides

The rate of change of the area can be found by taking the derivative of each side of the equation with respect to time:

The area of a circle is given by the function:

This equation can be rewritten to define the radius:

For the area function

The radius is then

or

The rate of change of the radius can be found by taking the derivative of each side of this equation with respect to time:

The sides of a square and its area are related via the function

Rewriting the equation in terms of its sides gives

For the area definition

The sides are then

or

The rate of change can be found by taking the derivative of the equation with respect to time:

A cube's volume is defined in terms of its sides as follows:

For sides defined as

The volume is

The rate of change can be found by taking the derivative of the function with respect to time

The surface area of a sphere is given by the function

For a radius defined as

The surface area equation becomes

The rate of change of the surface area can then be found by taking the derivative of the equation with respect to time:

The area of a circle is defined by its radius as follows:

In the case of the given function for the radius 

The area is thus

The rate of change can be found by taking the derivative with respect to time:

The area of a right triangle can be written in terms of its legs (the two shorter sides):

For sides  and , the area expression for this problem becomes:

To find where this area has its local maxima/minima, take the derivative with respect to time and set the new equation equal to zero:

At an earlier time , the derivative is postive, and at a later time , the derivative is negative, indicating that  corresponds to a maximum.