Prove “the length of the arc is proportional to the central angle subtended”

Question

Depending on individuals’ experiences, the following might not (and hope not) happen to you. However, it did happen to some (including me).

When I first studied the topic on “central angle + arc length (also the area of a sector but skipped)”, my teacher did not give a rigorous proof on it (probably because of the question stated below). He just pointed out that “you can see for yourself that the wider is the central angle, the longer is the corresponding arc”. We were made to believe that “the length of the arc is proportional to the central angle subtended”.

The first question I want to ask is:- what method can we use to prove the above, using high-school level language?

Now, for those who accept such finding by observation, how about if I say “see for yourself that the wider is the central angle, the longer is the corresponding chord”. This further implies “the length of the chord is proportional to the central angle subtended”.

The second question is:- If the last remark, based on the second observation, is not true (and in fact it is NOT), why should we believe that from the first observation is true?

Answer 1

In "A Course of Pure Mathematics" by GH Hardy - I have the 10th edition of 1960 to hand and Chapter VII p316 para 163 - he observes:

The theory of the trigonometrical functions $\cos x, \sin x, \tan x$ as usually presented in text-books of elementary trigonometry, rests on an unproven assumption. An angle is the configuration formed by two straight lines $OA, OP$; and there is no difficulty in translating this 'geometrical' definition into purely analytical terms. The assumption comes at the next stage, when it is assumed that angles are capable of measurement, that is to say that there is a real number $x$ associated with the configuration $\dots$

He then goes on to say that a normal approach is to define the angle by the length of an arc but

It has, however, for our present purpose, a fatal defect; for we have not proved that the arc of a curve, even of a circle, possesses a length.

And having treated integrals and areas already, proceeds to define the angle in terms of the area.

So your question goes to the heart of a foundational question which is not often noticed. If you want a more careful treatment, you really have to take full care - otherwise you are simply hiding the key issue at hand.

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Of course length proportional to angle can also be looked at by examining regular polygons inside and outside the circle and arguing from congruent triangles. This essentially involves creating a definition of arc length.

Answer 2

You know that the perimeter of the circle is $2\pi R$, with $R$ being the radius.

This is the lenght of the arc corresponding to angle $2\pi$.

The length of the arc corresponding to an angle of $\theta$, with $0<\theta<2\pi$ is:

$L=\theta R$

EDIT: of course this is a formula. However you can relatively easily see that half the angle leads to half the length, and also a quarter, for simple basic symmetry. Thus the proportionnality.

Answer 3

When you have two central angles $\alpha$, $\beta$, and their ratio is rational, then there is a very small angle $\gamma$ such that $\alpha=m\gamma$, $\beta=n\gamma$ with $m$, $n\in{\mathbb N}$. It follows from "elementary euclidean principles" (i.e. congruence, etc.) that $${\rm arc}(\alpha)=m\>{\rm arc}(\gamma),\quad {\rm arc}(\beta)=n\>{\rm arc}(\gamma)\ ,$$ which proves proportionality of arc lengths at least for rational angle ratios.

Answer 4

We know that,

If y is directly proportional to x, then the equation of the line would be $$y = mx$$ i.e. $$\frac{y_1}{y_2} = \frac{x_1}{x_2}$$ Now, if we take $$\theta_1 = 360^o$$ $$l_1 = 2\pi r$$ and, $$\theta_2 = 180^o$$ $$l_2 = \pi r$$ ( $\theta_1$ and $\theta_2$ correspond to the angle subtended by arcs of length $l_1$ and $l_2$ respectively )

We see that, $$\frac{\theta_1}{\theta_2} = \frac{l_1}{l_2}$$ which implies that central angle subtended is directly proportional to length of the arc.

Answer 5

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For This,I would like to consider asking,how do we define length of Arc of a circle,I would like to present one way to look at it,

So Assume two concentric circles,With common centre 'O',radii,$R_1$ and $R_2$ respectively

Draw two n-sided 'Regular' polygons on corresponding concentric circles with common centre 'O' ,join all vertices of polygons with centre of Circle

Using Similiarity,Or Basic proportionality theoram We can say that

$R_1/R_2$=$L_1/L_2$

Now I want you to visualise the same thing by Increasing the number of sides from 6,7,8,9.......till you get to a very large number,as we increase the number of sides,We get that the lengths $L_1$ and $L_2$ tend to equal The corresponding arc lengths,By increasing the number of sides furthermore(eventually,Mathematicians say they tend to infinity),the approximation gets good enough to Say length of each side equals to Corresponding Arc length

Now we can say $R_1/R_2=dL_1/dL_2$ where dL just denotes very small length of that arc

Now we can do this with every side of That regular polygon

So from here $R_1/R_2$=Sum of all such $dL_1$/sum of All such $dL_2$

And this Sum indeed equals Curcum ference of circle

From Here $R_1/R_2=C_1/C_2$ Rearranging $C_1/R_1=C_2/R_2$=$k(constant)$,This constant is called $2π$ Hence $C=2πR$

Now during this proof,due to regular polygon,The Angle subtended By each side can Be given As $2π(radian)/n$

From here,As polygon is regular,All sides are equal too,Also all angles equal 2π/n=∆ so for one side Let length be L1 and angle be ∆ Similarly for second side let length be L2(indeed it equals L1),so for L1+L2=2L1 the angle is 2∆

Hence,We conclude length of Arc is proportional to angle subtended

Hence L1/∆1=L2/∆2=k Hope You understand this