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Proof 3: Circle circumference and arc length

The circumference formula is quick to figure out since by definition: $\text{circumference}=\pi\;\text{diameter}$ (as explained in the Natural constants and pi skill) and since the diameter is double the radius $d=2r$:

$$L_\text{circumference}=\pi\,d=\pi2r\quad_\blacksquare$$

If we only have a section of a circle – an arc – then we must know the angle $\theta$ that this arc spans over. The portion – the ratio – of the circle that we have is $\frac{L_\text{arc}}{ L_\text{full round} }$ (if we have, say, 1 metre of a circle that is 4 metre around, then we have one fourth, $1/4$). the angle covers that same ratio $\frac \theta { \theta_\text{full round} }$ . These two ratios are the same:

$$\begin{align}
\frac{L_\text{arc}}{ L_\text{full round} } &= \frac \theta { \theta_\text{full round} }\quad \Leftrightarrow\\
L_\text{arc} &= \frac \theta { \theta_\text{full round} } L_\text{full round} \\
~&= \frac \theta { \theta_\text{full round} } ~2\pi r\qquad \leftarrow L_\text{full round}= L_\text{circumference}=2\pi r \\
~&= \frac \theta { \cancel{ 2\pi} } ~ \cancel{ 2\pi} r\qquad \leftarrow \text{A full round is } 2\pi\text{ radians} \\
~&= \theta r \quad _\blacksquare
\end{align}$$

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