A sector is a pie slice; an arc is the crust on that slice. No new machinery — just a fair share. The centre angle $\theta$ (the Greek letter theta) tells you the slice's share of the full $360^\circ$: a $90^\circ$ slice is $\tfrac{90}{360} = \tfrac{1}{4}$ of the pie, crust included.

Then take that fraction of whatever the whole circle owns. Slice of the area: $A = \dfrac{\theta}{360^\circ} \cdot \pi r^2$ — A equals theta over three-sixty, times pi r squared. Slice of the crust — that length is the arc, $s$: $s = \dfrac{\theta}{360^\circ} \cdot 2\pi r$ — s equals theta over three-sixty, times two pi r. Fraction first, then the familiar formula: two old friends, each wearing a slice-sized coat. The angles here all divide 360 cleanly — the fractions are friendly on purpose.