# What is the maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W?

Let us set up a concrete scenario...

Start with a rectangle with vertices:

So the maximum area is:

Unsurprisingly, this is when the circumscribing rectangle is a square.

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And now with rotations.

The circumscribed quadrilateral area is given by

so

Now the maximum is at the solution of

giving

so the solution is for

because at this point

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Now using the Lagrange Multipliers technique.

The circumscribed rectangle has the side dimensions

The restrictions are

The lagrangian is

The stationary points are the solutions of

or

Solving for

and the maximum area is

Of course the minimum area circumscribing rectangle has the area

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The maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W is achieved when the circumscribing rectangle has its vertices touching the midpoints of the sides of the given rectangle. In this scenario, the length of the circumscribing rectangle is equal to twice the length of the given rectangle (2L), and the width of the circumscribing rectangle is equal to twice the width of the given rectangle (2W). Therefore, the maximum area ( A_{\text{max}} ) of the circumscribing rectangle is given by the formula:

[ A_{\text{max}} = (2L) \times (2W) = 4LW ]

So, the maximum area of the circumscribing rectangle is four times the area of the given rectangle.

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