An arc of a circle is exactly as long as the radius of the circle…

Step 1. The equation. $\dfrac{\theta}{360} \times 2\pi r = r$.

Step 2. The radius cancels. Divide both sides by $r$: $\dfrac{\theta}{360} \times 2\pi = 1$. So the angle does not depend on the radius.

Step 3. Solve. We want $\theta$, so $\theta = \dfrac{360}{2\pi} = \dfrac{180}{\pi}$ degrees. With a calculator this is $57.3^\circ$.

Why it matters. In A-Level and IB this angle is the unit for angles. If $\theta$ is measured in radians, the arc is simply $l = r\theta$.