The Circumference, an Inexorable Geometric Figure

The circumference is a well-known two-dimensional figure and is part of people’s daily lives. The term circle is often used to mention it. Let it be clear: circle and circumference are different geometric figures. Circumferences are generally approached with the implicit idea that there are circles there. The circumference is a curved and closed line, unlike the circle which is a region delimited by the circumference and including it.

We will approach the circumference here in a way that we will try to avoid involving the famous mathematical constant π, since there are exhaustive known and available approaches to this. The idea is to contemplate the properties of the circumference from a different point of view and using the language of plane geometry.

1️⃣Zoom in on Circumference

In a plane, a circumference is said to be the locus of points equidistant from a central point called the center of the circumference.

Let OD be a line segment whose measure

is equal to a R value. And let’s assume a circle

whose points are distant R from the central point O. This distance R is called the radius of the circle. The outline of this circumference is shown in figure 1

Figure 1. Drawing a circumference.

Given any two points on a circumference, there are two paths between these two points to consider. One corresponds to the line segment that joins them, called a chord, and the other, which corresponds to the stretch of circumference delimited by them, called an arc. The distance between such points passing through the arc is the arc length.

Let t be the length of the largest arc of the circumference, that is, itself. Let’s assume that there is a value κ such that:

At first there is a symmetry that impels us to approach any part of the circumference indistinctly. Then we divide it into n parts such that each part covers equal regions of the underlying circle of the circumference. This can be done by drawing inscribed regular polygons. As an illustration, such a geometric construction is shown in figure 2 , with the pentagon as the “pivot’’ polygon.

Figure 2. Circumference fractionation using a regular polygon as a template.

Let us now imagine that we are tracing the circumference T already on the verge of taking the last step of the n steps that we divide it. This step corresponds, for example, to a segment DC as shown in figure 2. Even if the previous steps, by hypothesis, are exactly corresponding to the predestined geometric locus, this last step will be an exception, since it is a line segment. And so counting, we will have as length:

or

And this way,

In this case, since we accounted for the last step, we were unable to build the circumference as a whole. There is an operational limitation, although the geometric loci are defined. A wire of known length to be bent in the form of a circumferencewill have an “apparent” radius R′, that is,

then

If, on the other hand, we had a sufficient amount of wire to construct a circle of known radius, everything leads to the belief that we would not be able to foresee the necessary quantity.

If we had to sum it up in words, we would say: the circumference is a place with a declared address, but with a challenging location.

2️⃣ Focusing on “Genetics”

On the chord DC we will now make a second geometric construction, first we identify the midpoint of this chord, D′ , and then using our paradigm we will draw a new circle with center in that midpoint and radius r. This construction is depicted in figure 3, where their respective adjacent circumferences will also appear for the other polygonal elements. For the sake of visual simplification, only the semicircumferences were represented.

Figure 3. Caminhos circunferenciais adjacentes ao lados do polígono inscrito em T.

In the same way, let’s do the situation of the circumference S

On the other hand, we say that the proportion

and so we can do

multiplying this expression by 2n, we get

or

The factor 2n is justified by the bisector effect taken by the support line of the apothem OD′ when dividing the circumference S in half. If we make n as large as possible, what will we represent as

we will have the expression

on which we deem relevant the following considerations:

✍️ left-hand side of expression: as a matter of fact this term is related to the theoretical length t since we can simply do

and the term in parentheses will correspond to the sum of the sides of the polygon inscribed in T, that is,

and this sum corresponds in fact to the value t.

✍️ right-hand side: hat in the same way we will make it a multiple of κ,

or

Retro substituting the respective transformed expressions, we have

which, simplifying, we have the expression

It is worth noting that this identity has at its core the genetics of the distortion mentioned above, since, by the expression, it is not guaranteed that the parameters involved do not correspond to the last step mentioned above.

3️⃣ Risking a Value for κ

This doesn’t seem like a difficult task, using a measuring tape and pieces of yarn. But we will try another path, perhaps less precise and less laborious.

Let’s use a template, the square inscribed in the circumference. Its side will be equal to

and perimeter

In this way, we have the inequality

On the other hand, using the circumscribed square as a template, the length of its side will be

and its perimeter

and so we have the inequality

So it is guaranteed that

This domain interval leads us to examine the trend of these values ​​and we will do this through the average value. Doing this, we have

or

To be a little more precise, let’s refine the methodology with two steps, namely:

• trim the corners of the circumscribed square to an octagon;
• transform the inscribed square into an octagon.

Graphic details are shown in figure 4.

Figure 4. Schematic to refine the process of searching for the value of κ.

For the circumscribed octagon its perimeter will be

or

For the inscribed octagon its perimeter will be

or

Then the average value sought will be

that simplifying

Continuing this methodology, everything leads to believe that the next value may be more precise and would probably be a real number multiplied by R, that is,

where this ε is a value resulting from a methodology or algorithm.

We could, without error, write:

4️⃣ Discovering the Wheel

In describing these ideas we are not “rediscovering the wheel”. Perhaps we are prospecting in already explored soil. However, there is always something to be found that went unnoticed. In this case there is no denying that there are two things that we can highlight
- the circumference as a geometric locus;
- the circumference in the sense of their related magnitudes.
Bending a line segment to form a circle creates an intrinsic measurement uncertainty.

This is the circumference, this inexorable geometric figure!