📂Geometry

Proof that the Length of an Arc and the Length of a Chord are Approximately Equal When the Central Angle is Small

Theorem

When the central angle $\theta$ is sufficiently small, the length of the chord and the length of the arc approximate each other. When $\theta \rightarrow 0$,

$$\overline{AB} \approx \stackrel\frown{AB}$$

Proof

In the figure above, the length of the chord is

$$\overline{AB} =2\overline{AM}=2r\sin \frac{\theta}{2}$$

The length of the arc with a central angle of $\theta$ and the radius length of $r$ is

$$\stackrel\frown{AB}=r\theta$$

When the angle is sufficiently small, saying the length of the arc and the length of the chord approximate each other means that the difference between them is almost negligible, i.e., the ratio is 1,

$$\lim \limits_{\theta \rightarrow 0}\dfrac{ \overline{AB} }{\stackrel\frown{AB}}=1$$

should be confirmed.

$$\begin{align*} \lim \limits_{\theta \rightarrow 0}\dfrac{ \overline{AB} }{\stackrel\frown{AB}} =&\ \lim \limits_{\theta \rightarrow 0} \dfrac{2r\sin \frac{\theta}{2}} {r\theta} \\ =&\ \lim \limits_{\theta \rightarrow 0} \dfrac{\sin \frac{\theta}{2}}{\frac{\theta}{2} } = 1 \end{align*}$$

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