Finding the Largest Inscribed Rectangle


THE PROBLEM: What is the area of the largest rectangle which can be inscribed in a circle of radius 1?


Set-up

We are asked to consider a fixed circle and all rectangles which can be inscribed in the circle. Among those there is, presumably, one whose area is largest. We are asked to find this area, but we may also have to identify the rectangle which achieves this area along the way.

To make a mathematical model we draw a diagram and label its parts. We note that the radius of the circle is constant and that all parameters of the inscribed rectangle are variable.

The quantity we need to maximize is the area of the rectangle which is given by

A = wh.

We note that w and h must be non-negative and can be at most 2 since the rectangle must fit into the circle.

At this point we have TWO variables and need to eliminate one of them. To do this we need to find a relation between the width and the height. This relation must come from the fact that the rectangle is inscribed in the circle, a fact which is central to the problem and therefore must be used.

After considering various possibilities, we draw a line segment whose length is labelled x. This gives us a right triangle with sides x and y. We then note the relationships:

x = w/2     y = h/2     x2 + y2 = 1.

We are now ready to do most of the necessary algebra.


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© CalculusQuestTM
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All rights reserved---1996
William A. Bogley
Robby Robson