Dr. Sarah's Maple Demo

A Circle and Rectangle Can Never Have Both the Same Area and Perimeter

A Circle and a Square Can Never Have Both the Same Area and Perimeter

To see that a circle and a square can never have both the same area and perimeter, we simultaneously solve

these conditions for the radius of the circle, r, and the side length of the square, x. Execute the following

Maple command:

> solve({x^2=Pi*r^2,4*x=2*Pi*r},{x,r});

[Maple Math]

Notice that Maple outputs one solution.

Why doesn't the solution give us a circle and square with equal area and perimeter. What does the solution look like geometrically?

Now we will investigate why a circle and rectangle can never both have the same area and perimeter.

Algebraic Explorations for Circles and Rectangles

Let r be the radius of a circle, and x and y be the sides of a rectangle. If the area of the circle equals the area of the rectangle, then we can solve for the condition on r that this imposes:

> solve({x*y=Pi*r^2},{r});

[Maple Math]

If the perimeter of the circle equals the perimeter of the rectangle, then we can solve for the condition on r that this imposes:

> solve({2*x+2*y=2*Pi*r},{r});

[Maple Math]

Since the radius of a circle can never be negative, using algebra, we see that x+y must be equal to [Maple Math]

Can you solve for x and y by hand? Explain.

> solve(x+y=sqrt(x*y*Pi),{x,y});

[Maple Math]

Notice that Maple uses I to represent [Maple Math]

If y is non-zero, why can't x ever be real? Explain.

If y is 0, then x can be real. What is x and r in this case? Why doesn't this real solution yield a circle and a rectangle with both have the same area and perimeter.

Graphical Explorations for Circle and Rectangle

As above, if the perimeters and areas match up then we know that x+y equals [Maple Math]

We'll now look at a 3d graph of the surfaces z=x+y and z= [Maple Math]

with x values ranging from 1 to 4 and y values ranging from 1 to 4. Note that r=z, and so if we find an intersection point to the two surfaces, then that tells us that we have a solution for the radius r.

> plot3d({x+y,sqrt(x*y*Pi)}, x=1..4, y=1..4);

On first glance it appears that these surfaces do intersect. Here is the same plot from another viewpoint to try and get a better view of any intersections.

It still appears that these surfaces intersect.
Click on the graph in Maple and change the view to see that there is no intersection.