AI Tutor
Imagine a bakery serves you a single slice of circular pie whose outer crust measures exactly 4π cm4\pi\text{ cm}, but the baker forgot to mention how many equal slices the pie was cut into.
How do you work backward from that curved edge to reveal the exact opening angle at the center of the pie?

The Part-to-Whole Ratio

A full circular turn measures 360∘360^\circ, with a total outer boundary called the circumference, given by C=2Ο€rC = 2\pi r.
An arc is just a fraction of that boundary, carved out by a central angle (labeled ΞΈ\theta) whose vertex sits right at the circle's center.
Because the arc is the exact same fraction of the circumference as the angle is of a full circle, our core equation is ΞΈ360∘=s2Ο€r\frac{\theta}{360^\circ} = \frac{s}{2\pi r}, where ss is the arc length.
This relationship is the Fundamental Proportionality Principle: an arc length ss and sector area AA represent the exact same fraction of a complete circle's circumference (C=2Ο€rC = 2\pi r) and area (Acircle=Ο€r2A_{\text{circle}} = \pi r^2).
That fraction is set by the central angle ratio: ΞΈ360∘\frac{\theta}{360^\circ} for angles measured in degrees, or ΞΈ2Ο€\frac{\theta}{2\pi} for angles measured in radians.
What happens if we rearrange this relationship so the unknown angle ΞΈ\theta sits alone on one side?

Solving for the Central Angle from Arc Length

Multiplying both sides of the ratio by 360∘360^\circ isolates our central angle: ΞΈ=s2Ο€rΓ—360∘\theta = \frac{s}{2\pi r} \times 360^\circ.
Let's put this into action: a wheel with a radius of r=10Β cmr = 10\text{ cm} rolls across a desk, tracing an arc of s=5π cms = 5\pi\text{ cm} along its rim.