# 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.

By signing up, you agree to our Terms of Service and Privacy Policy

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

By signing up, you agree to our Terms of Service and Privacy Policy

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

By signing up, you agree to our Terms of Service and Privacy Policy

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.

By signing up, you agree to our Terms of Service and Privacy Policy

- How do you use linear approximation to estimate #root3( 64.1)#?
- How do you use Newton's method to find the approximate solution to the equation #tanx=e^x, 0<x<pi/2#?
- A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is minimum?
- What dimensions would you need to make a glass cage with maximum volume if you have a piece of glass that is 14" by 72"?
- How do you find the point on the curve #y=2x^2# closest to (2,1)?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7