The Circled Square Math Problem Claims Pi Equals Four


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Diameter of a Circle in the Perimeter of a Square : Image by Mike DeHaan

In the spring of 2013, an Internet “troll” cartoon attracted some attention with a math problem stating that “pi equals four”. This Yahoo question was posted in December 2010 at Shouldn’t Pi be equal to 4? by skibm80, and the puzzle may be older than that.

Of course, “pi=π=3.14159276…” approximately, so this raises a number of questions:

  • What did the troll claim in his “circled square” math problem?
  • Can we disprove this claim by measuring the circumference of a circle?
  • Did the troll make an incorrect argument in posing his math puzzle?
  • Can we make an estimate of pi by iteration?

The Circled Square Math Problem in Cartoon Form

The first frame of the cartoon shows a unit circle inside a unit square. The diameter of the circle equals the length of each side of the square. Therefore the diameter of the circle equals the length of each side: “d=s=1”.

The perimeter of the square is “P=4*s=4*d=4”, and the circumference of the circle is “C=π*d=3.14…”.

The area of the square is “AS=d*d=1″.

The area of the circle is “AC=π*((d/2)2)=”π*(1/2)*(1/2)=π/4=0.785…”.

In the next frame, each corner is “cut out” of the square. By folding each corner over the diagonal, each apex of the square’s corner now touches the circle.

Clearly the area of the new figure is less than the original square. However, the perimeter of the new figure is the same.

Repeat this process by folding ever-smaller corners so each prior apex, the point farthest from the circle, next touches the circle.

The area of each new figure shrinks, and approaches the value of the circle.

However, the perimeter of each new figure is always 4.

Begin Collapsing the Square toward the Circle : Image by Mike DeHaan

The troll finally claims that this proves that “π=4”, since the new figure always becomes a closer approximation of a circle, yet the length of its perimeter remains four.

Disproof by Measuring the Circumference of a Circle

The most obvious disproof of the troll’s circled square math problem’s result is to directly measure the circumference of the circle. Roll Out the Simplest Mathematics Activity for Pi Day described a very straightforward way to measure a circle and calculate pi.

Clearly, the value for pi is more than 3 but much less than 4, as shown by direct measurement.

Flaws in the Argument for the Circled Square Math Problem

The troll’s argument correctly states that the perimeter of each new figure remains 4 units. As well, the path taken by the new perimeter is closer to the circle, and therefore becomes a better approximation of a circle.

Also, the area in the new figure becomes closer to the area of the circle.

The Square Collapses Further toward the Circle : Image by Mike DeHaan

One might think this implies that the circumference of the circle must be the same as the perimeter of the square. After all, the new figures are becoming more circular with each iteration.

Things of Interest: Troll Pi notes that “the limit of a sequence isn’t necessarily a member of that sequence”. The succession of approximations is not necessarily the same as the final value.

Note that the limit value does not change as the perimeter changes shape. Normally one expects that successive approximations will make the value more nearly correct from one iteration to the next. (The final section of this article demonstrates improvement by iteration).

Since the perimeter has a constant value of four, there is clearly some discontinuity between any new figure’s perimeter and a circle’s circumference.

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