### Classroom Activity Sheet on
Using Algebra to Show that a Circle and a
Square Can Never have Both the Same Area and Perimeter

We'll show that a circle and a square have never have both the same
area and perimeter.
If r is the radius of a circle
then the area of the circle is Pi r^2 and the circumference of the
circle is 2 Pi r.
If x is the length of the side
of a square, then
the area of the square is x^2 and the perimeter of the square is 4x.

1) For Equation 1, set the area of the circle equal to the area of the
square.

2) For Equation 2, set the circumference of the circle equal to the
perimeter of the square.

3) Solve this system of two equations. There is one solution - what is it?
What does this solution mean geometrically?

4) Can a circle and square have the same area? If so, then give
an example. If not, then give a proof.

5) Can a circle and square have the same perimeter? If so, then give
an example. If not, then give a proof.