Geometrical Constants - Part 3

5 min read
Geometrical Constants - Part 3

In the previous part, we extended the idea of the geometrical constant $\mathcal{C}$ to a square and showed that the values of $\mathcal{C}$ increases with the number of sides of the polygon. Towards the end of the post the readers were left with two questions, and this final part of the series will answer them both.

The magic formula for an $n-$sided polygon

Previously we saw the value of $\mathcal{C}$ for a few cases without calculations, and it is natural to expect a formula that can give us the value for any polygon. If we understand the basic idea behind the calculations, it becomes just a small leap of imagination for an $n-$sided polygon.

Let us try to see a pattern in those calculations with a diagram -

Here is what we observe:

  1. The angle subtended by the sides at the center is different for each polygon.
  2. As the number of sides increases, the value of this angle decreases.

What does all this mean? Well, it means that there is a connection between the number of sides and the angle subtended. Let us explore this connection with the help of the following diagram -

We can clearly see that there are $n$ sides in this polygon. It may not seem like a perfectly symmetric polygon because we are trying to fit an arbitrary number of sides, but I hope you get the idea. It is also clearly shown that the complete angle (shown in blue) about the central point is obviously $360^{\circ}$. So, the angle subtended by each of the sides (shown in red) must be $\displaystyle \frac{360^{\circ}}{n}$. That is because, there are $n$ such angles which are equal in size due to symmetry, so the sum of all these angles becomes $\displaystyle n \times \frac{360^{\circ}}{n} = 360^{\circ}$, as expected.

We can now study one of triangles of the polygon in the following way -

The angle $\angle AOD$ is half of angle $\angle AOB$ because the the line segment $OD$ divides the angle $\angle AOB$ as well as the side $AB$ into two equal parts due to symmetry (more technical reason: It is an Isosceles triangle). So, we can write

$$\angle AOD = \frac{1}{2} \times \frac{360^{\circ}}{n} = \frac{180^{\circ}}{n}$$

Now, we use a basic trigonometric formula: $\displaystyle \sin{\theta} = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$ for the triangle $AOD$, to obtain

$$\sin{\left(\frac{180^{\circ}}{n}\right)} = \frac{AD}{AO} = \frac{AD}{r}$$
so that we can write
$$AD = r\sin{\left(\frac{180^{\circ}}{n}\right)}$$

From the diagram we can see that the side of the polygon is given by $AB$ which is twice of $AD$ (due to symmetry). Therefore, the length of the side of the polygon becomes

$$AB = 2\times AD = 2r\sin{\left(\frac{180^{\circ}}{n}\right)}$$

If the side of the polygon is known, it becomes trivial to find the perimeter, which is just the sum of all the sides. Since there are $n$ sides in this case, the formula for perimeter becomes

$$\text{Perimeter} = n \times AB = 2nr\sin{\left(\frac{180^{\circ}}{n}\right)}$$

Finally, if the perimeter is known, we can obtain the value of the geometrical constant $\mathcal{C}$ as

$$\begin{aligned}\mathcal{C} = \frac{\text{Perimeter}}{OA} &= \frac{2n\cancel{r}\sin{\left(\frac{180^{\circ}}{n}\right)}}{\cancel{r}}\\ &= 2n\sin{\left(\frac{180^{\circ}}{n}\right)}\end{aligned}$$

Thats the final result! This is the magic formula for $\mathcal{C}$ that works for any polygon. The reader is encouraged to test this formula by choosing different values of $n$ and compare the results with the known cases in the previous two articles.

Let us now come to the last piece of the puzzle.

What is so special about the number $6.282 \cdots$ ?

To answer this question, we need to raise the level of mathematics just a little bit. Let us start by stating a very powerful mathematical fact -

If the value of any angle $\theta$ is very small, then $\sin{\theta}$ can be replaced by just $\theta$. That means $\sin{\theta} \rightarrow \theta$, when $\theta \rightarrow 0$

This is a rather technical statement, and we shall not prove this here. For the purpose and scope of this article, you have to trust me on this without proof.

(If you still want to know about it, there is another article where a complete proof is discussed.)

There is one more thing that we need - So far, we have written our angles in degrees. In order to use the above statement, we must convert the angles into another units called radians.

But why do we need to do it? Well, it turns out that radian is sort of the more "natural" unit for angles (check out the proof for details). So, here is the conversion factor for radians -

$$180 \text{ degrees} = \pi \text{ radians}$$

Thats it! This is all we need to proceed, and we can rewrite our magic formula in terms of radians as

$$\mathcal{C} = 2n\sin{\left(\frac{180^{\circ}}{n}\right)} = 2n\sin{\left(\frac{\pi}{n}\right)}$$

Now, if we ask an important question -

What will happen if the polygon has too many sides?

Mathematically speaking, we want to know what will happen if $n$ becomes too large. Well, we can visualize that a polygon with too many sides looks very much like a circle!

We also know that it we divide a positive number by a very large number, the result is a very small number, right?
So, if $n$ becomes a large number, it would mean that $\displaystyle \frac{\pi}{n}$ becomes very small. Therefore, we can use the above stated fact to write

$$\sin{\left(\frac{\pi}{n}\right)} = \frac{\pi}{n}.$$

Finally, using this result we can write the magic formula obtained in the previous section as

$$\mathcal{C} = 2n\sin{\left(\frac{\pi}{n}\right)} = 2n \times \frac{\pi}{n} = 2\pi$$

Let us a take a moment to realize what a remarkable result this is!

We have proved that the ratio of the perimeter to the radius of any circle is a constant and that value happens to be $2\pi$. Most of us learnt in school that the circumference (perimeter) of a circle is given by the formula $C = 2\pi r$, which means $\displaystyle \frac{C}{r} = 2\pi$. So what we have really done is systematically derived this very famous result, while learning a lot about shapes which have nothing to do at all with circles!

And, if we plug in the value of $\pi = 3.141 \cdots$, we can see that $2\pi = 6.283 \cdots$. Thats the mysterious number that we were trying to explain!

So, all this time, we have been chasing down the origin story of $\pi$. And now we know how and why this is related to symmetry of polygons. After all, A circle if the most symmetric polygon there is!

Geometrical Constants - Part 2
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Geometrical Constants - Part 2

In the previous part, we obtained a geometrical constant for an equilateral triangle, and its value was approximately 5.196. In the second part, we will continue the calculations..

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