The task is to express the length of an arc of a circle trapped between two radii named $r$ if the angle between them is infinitesimally small, named $d\theta$.
The solution to this problem is supposed to be:
$$l=r \cdot {d\theta}$$
but I do not understand why this would be the case.
I've provided a simple Paint sketch for more details:
Since $d\theta$ is infinitesimally small, line $\overline{BE}$ is infinitesimally small as well, so we note that with $\overline {BE} = dx$. Since $dx$ is so small, its projection $\overline {BD}$ and circular chord $\overline{BC}$ can be approximated as:
$$\overline{BD} \approx \overline{BC} \approx dx$$
The similar can be concluded for the radii:
$$\overline{AB} = \overline{AC} = r$$
$$r \approx \overline{AD} \approx \overline {AE}$$
The task was to find the length of an arc limited with the chord $\overline {BC}$, that is, trapped between $\overline{AB}$ and $\overline{AC}$. The solution should be $l=r \cdot {d\theta}$, but how?
Formula for deducing arc length is:
$$l=2r\pi \frac{d\theta}{360°} = r\pi \frac{d\theta}{180°}$$
The only way for this to become $l=r\cdot d\theta$ is if $\pi$ and $180°$ would somehow cancel each other out. $\pi rad$ indeed has the value of $180°$, but this $\pi$ has the meaning of length, that is $\pi \approx 3.14$, and not radians.
If there I did not specify something enough, please let me know, so I can explain myself better. Thank you in advance.
Best Answer
Consider a circle has radius $r$ units. length of circumference is $2\pi r$ which produces total angle $2 \pi$ radians. unit radian angle produce arc length$= \frac{2\pi r}{2\pi} = r$ units "$\theta$ radian (no matter how small or big it is )" angle produces arc length $= \theta r$.
ANOTHER WAY
Two end points of the arc is very close to each other forming an angle $dA$ in radians. For small angle $\tan dA = dA$. you can also draw a tangent at that point.
For right-angle triangle: $$\tan dA = dA = \text{length of arc / radius} $$
so length of arc $= r dA$ ...