Discover the Pi Paradox: The Infinite Perimeter of a Circle with a Diameter of 1

[jamester234]

Suppose you have a circle with a diameter of 1. If you draw a square with all four sides touching the circle, the perimeter of the square is 4. Now suppose you indent each corner of the square so that they all touch the circle-this will make a cross shape, and the perimeter of it is still 4. Now suppose you indent each corner of the cross so that they again all touch the circle. You can see now that the square is becoming more like a circle, and yet the pereimeter is still 4. This can be done to infinity, therefore, pi equals 4!

 

[Darken-Sol]

Suppose you have a circle with a diameter of 1. If you draw a square with all four sides touching the circle, the perimeter of the square is 4. Now suppose you indent each corner of the square so that they all touch the circle-this will make a cross shape, and the perimeter of it is still 4. Now suppose you indent each corner of the cross so that they again all touch the circle. You can see now that the square is becoming more like a circle, and yet the pereimeter is still 4. This can be done to infinity, therefore, pi equals 4!

the perimeter of your square would become roughly equal to the circumference if your circle. you still got some math to do before you get pi.

 

[g_edgar]

Nope. You have not proved the length of the circle (your limiting curve) is 4. You have proved, instead, that the length of the limit is not equal to the limit of the lengths. No need to use a circle to prove this: For example, you can use stairsteps converging to a sloping line segment.

 

[Stephen Tashi]

That demonstration is obviously wrong because it is contradicted by the proof that \pi = 2. Consider a circle of diameter 1 with its center at the origin and look at the part of it that lies in the 1st quadrant. Divide up the area under the curve of the circle into small rectangles in the manner that people do when they approximate integrals. The tops of these rectangles become closer and closer to the curve as their widths approach zero. The total of the lengths of the top of the rectangles is always 1/2 no matter how small they are. So the total perimeter of the circle is 2 when we sum over all 4 quadrants.

 

[Unrest]

This exact problem was just thrashed to death in another math forum on here.