What is an Angle?


7 min read
What is an Angle?

Introduction

The idea of an angle is one of the most fundamental in geometry. It comes naturally while studying properties of triangles where we establish the connection between its sides and angles. These connections are systematically constructed in the subject called Trigonometry.

What is an angle and how do we measure it?

Although the idea of an angle is visually quite intuitive, its mathematical definition is not so straightforward. Let us try to develop a visual method to understand how we define an angle.

Consider the following diagram

The "spread" between the lines $OA$ and $OB$ is what we might call an angle. Even if we choose the lines $OC$ and $OD$, the spread between the lines is the same. So, the choice of the pair of points is completely arbitrary.

Next, we embed this angle in a regular(symmetric) pentagon shown above. We can easily see that there would be 5 such angles subtended at the center by 5 sides respectively. The angles would all be equal because the sides are equal. We should keep in mind that we don't yet know what an angle is, but we can still establish some of its properties by thinking logically.

Let us call this angle $\theta$ (pronounced as "Theta"). Now, from the discussion above, its easy to say that the measure of $\theta$ should be one-fifth of the measure of the entire angle around the pentagon; thats just due to symmetry.

Let us take the measure of complete angle around the pentagon to be $\Omega$ (pronounced as "Omega"). Therefore, we can write

$$\theta = \frac{\Omega}{5}$$

Before we proceed, we need to state a very fundamental geometrical fact -

Measure of the complete angle is independent of the size and the number of sides of a polygon.

It seems fairly obvious that a bigger pentagon(or any other polygon) will have the same measure of the complete angle around the center. This means that $\Omega$ is some kind of a geometrical constant.

If we take a regular hexagon(6 sides) around the same angle $\theta$, the measure of the angle would turn out to be

$$\theta = \frac{\Omega}{6}$$

It looks as if the measure of an angle depends on the shape of the polygon around it. This is obviously unacceptable because an angle should have nothing to do with pentagons or hexagons or anything else. They are simply "spreads" between two lines. So we would like to come up with a better way to quantify angles.

Inspired by the fact that even for different shapes, the angle around the center is always $\Omega$, we could try to construct a quantity that connects $\Omega$ with a generic regular polygon. If we are lucky, this could lead to some sort of universality that we want. We begin by following the steps as described in another article to write a formula for the perimeter of an $n-$sided polygon as

$$\mathcal{P} = OA \times 2n \sin{\left(\frac{\Omega}{2n}\right)}$$

Now we use the plain old unitary method as follows

  • If we measure the perimeter $\mathcal{P}$ of any polygon, the angle subtended at the center is $\Omega$.
  • If we measure a unit length of the side of this polygon, the angle subtended must be $\displaystyle \frac{\Omega}{\mathcal{P}}$. We call this the unit angle.
  • For any other length $AB$ (diagram), the angle subtended $(\theta)$ must be $AB \times$ unit angle.

Using the expression of $\mathcal{P}$, we can write

$$\begin{aligned}\theta &= AB \times \frac{\Omega}{\mathcal{P}} \\ &= \frac{AB}{OA} \times \frac{\Omega}{2n \sin{\left(\frac{\Omega}{2n}\right)}}\\ &= \frac{AB}{OA} \times \frac{\Omega/2n}{\sin{\left(\frac{\Omega}{2n}\right)}} \end{aligned}$$

This expression looks really complicated, but I promise, we are going to make great simplifications.

Let us ask a question - What will happen if the value of $n$ is too large? First, the polygon will have too many sides, which will begin to look like a circle. Secondly, the value of $1/n$ will be very small. To make the expression a bit cleaner, we replace $1/n$ with $m$ and we understand that if $n$ becomes too large, then $m$ becomes too small. In terms of $m$, the last expression becomes

$$\begin{aligned}\theta &= \frac{AB}{OA} \times \frac{m\Omega/2}{\sin{\left(\frac{m\Omega}{2}\right)}} \end{aligned}$$

Now, we need to know what happens to ${\sin{\left(\frac{m\Omega}{2}\right)}}$ when $m$ is very small. To understand this, consider a triangle $\Delta ABC$ -

By definition, we know that

$$\sin{\alpha} = \frac{Perpendicular}{Hypotenuse} = \frac{BC}{AC} $$

So what will happen if we go from point $C$ to $C^{\prime}$ to $C^{\prime \prime}$ to $B$? The length of $BC$ is getting smaller and so is the angle $\angle A$. If the $\angle A$ becomes nearly zero, the length of $BC$ also becomes nearly zero. Let us rearrange the equation above, to get

$$BC = AC \times \sin{\alpha}$$

If we say that $BC$ is becoming zero, then we have

$$0 = AC \times \sin{\alpha}~~~~~ \Longrightarrow ~~~~~ \sin{\alpha} = 0$$

We have already established that if $BC$ becomes zero, then $\alpha$ also becomes zero. So we can write

$$\sin{\alpha} \to 0 \text{ when } \alpha \to 0.$$

Let us copy the expression for $\theta$ from above -

$$\begin{aligned}\theta &= \frac{AB}{OA} \times \frac{m\Omega/2}{\sin{\left(\frac{m\Omega}{2}\right)}} \end{aligned}$$

In our case, $m$ is becoming nearly zero. Now, any number multiplied with $m$ will also become zero, so that $m\Omega/2$ is a really small number. By following the arguments discussed above, this should mean ${\sin{\left(\frac{m\Omega}{2}\right)}} \to 0$ when $m \to 0$. Although this is quite a nice bit of simplification, but this a disaster! It means that we are dividing a quantity by zero, which is mathematically illegal. Does this mean, we made an error? No, we didn't.

Let us identify the source of this apparent paradox. There is a quantity in the denominator which takes a value equal to zero for a certain case. The only way to get rid of this problem is to cancel this term with something in the numerator, so that when $m$ becomes zero, there is no term left in the denominator to cause any damage. Now, the only way it can happen is if we have something like -

$$\sin{\left(\frac{m\Omega}{2}\right)} \to m \times k, \text{ if } m \to 0$$

The number $k$ is a constant, and its sudden appearance might be confusing. A mathematically rigorous explanation of $k$ is beyond the scope of the article. At this moment, all we can say is that on the right hand side of the equation, we must have a factor of $m$ multiplied with some unknown object. This object in principle could be anything, but to make our analysis simple, we assume it to be a number. We will worry about this number in the next section, but the most important thing is that the $m$ now cancels with the numerator, and the division by zero problem is resolved!

This may look like cheating, but it is a legit mathematical technique. Besides, its the only sensible way to get out of the division-by-zero issue. In fact, this in itself is a very fundamental relation, with huge implications in another branch of mathematics called Calculus.

Upon this replacement, the expression for $\theta$ becomes

$$\begin{aligned}\theta &= \frac{AB}{OA} \times \frac{m\Omega/2}{m\times k} \\ &= \frac{AB}{OA} \times \frac{\Omega}{2k}\end{aligned} $$

As we can see, the zero in the denominator just vanished!

We already know that when $m \to 0$, we are dealing with a circle and not a polygon (diagram). We may identify $OA$ as the radius of the circle and $AB$ as a circular arc of a certain length depending on $\theta$.

But how do we find the value of $k$?

To answer to this question, we will use the oldest trick in the book - physical measurement. Let us look at the following diagram -

Here we have taken $\theta$ as half of the complete angle around the circle. So its measure is $\displaystyle \Omega/2$. If we use our definition of an angle in this case, we will get

$$\theta = \frac{\Omega}{2} = \frac{\text{arclen}(AB)}{OA} \times \frac{\Omega}{2k}$$

The $\Omega$ gets cancelled on both sides, and we are left with a simple equation -

$$k = \frac{\text{arclen}(AB)}{OA}$$

If we can physically measure the length of the semi-circular arc for a given value of radius, we can plug those values in this formula and obtain the numerical value of $k$. It turns out that the value of $k$ is approximately $3.14$ for a circle of any size. In mathematical literature, this value is called $\pi$ (pronounced as "pi"). Therefore, we can use this value of $k$ in our definition of the angle to get

$$\begin{aligned}\theta &= \frac{\text{arclen}(AB)}{\text{Radius}} \times \frac{\Omega}{2\pi}\end{aligned}$$

So far we have been avoiding to talk about the value of $\Omega$, because its a completely arbitrary choice. It is something like measuring length of a rod in metric system vs imperial system. The actual length of the rod does not change, but the numbers associated might change. So, we are completely free to choose a value for $\Omega$ in a particular system. By looking at the last formula, it makes a lot of sense to simply choose $\Omega = 2\pi$. This choice makes the definition of the angle really simple -

$$\theta = \frac{\text{arclen}(AB)}{\text{Radius}}$$

Angles measured with this choice of $\Omega$ are called radians. If we choose $\Omega = 360$, we call them degrees.

To summarize:

  1. We have found an unambiguous definition of an angle(measured in radians), i.e. Angle = Arc-length/Radius.
  2. If we are measuring angles in radians, we have a very useful result: $\displaystyle \sin{\theta} \to \theta$ when $\theta \to 0$.

We will use the second point in a future article.

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