Maximal area quadrilateral Show that among all the 4sided po

Maximal area quadrilateral. Show that among all the 4-sided polygons inscribed in a circle (see the diagram for an example) the squares have the largest area.

Solution

Here we have to prove that the square has the greatest area for a given/fixed perimeter when a rectangle is considered inscribed in a given circle.

So the perimeter P can be taken to be a constant P.

Now let us use a variable l for the length. So the width will be given as P/2 - l.

The area of a rectangle is given by

A=l*b ; (length*width)

The area A = ( P/2 -l) *l

Now to maximize the area, for a variable l, we have to find dA/dl.

dA/dl= P/2 - 2l.

For a maximum ,the slope dA/dl=0.

thus Equating this to zero we get

P/2 - 2l=0

=> l= P/4

So the length of the rectangle will be P/4 and the width will be P/2 - l= P/4.

Hence length =width

this is condition of a square.

Therefore the area of a given quadilateral is the largest if the shape is that of a square with each side equal to the perimeter divided by 4.