AI Tutor
If you eat half a 12-inch pizza, you eat half of the total crust along the rim. If you eat a single slice with a 60∘60^\circ tip, what exact length of crust do you get?
Because a circle is symmetrical, any slice's curved edge is the exact same fraction of the whole perimeter as its central angle is of a full 360∘360^\circ rotation.

Parts of the Whole

The distance all the way around a circle is its circumference (C=2πrC = 2\pi r), where rr is the radius—the distance from the center to any point on the edge.
An arc length (often written as ss or LL) is the linear distance along a curved portion of the circle, measured in standard length units like inches or centimeters.
Around 240 BCE, the Greek scholar Eratosthenes used this exact ratio to calculate the entire circumference of the Earth using only the angle of the Sun's shadow between two cities.
We express this relationship as a direct proportion:
Arc LengthCircumference=Central Angle360∘  ⟹  s2πr=θ360∘\frac{\text{Arc Length}}{\text{Circumference}} = \frac{\text{Central Angle}}{360^\circ} \implies \frac{s}{2\pi r} = \frac{\theta}{360^\circ}
Multiplying both sides by 2πr2\pi r gives the standard arc length formula: s=(θ360∘)⋅2πrs = \left(\frac{\theta}{360^\circ}\right) \cdot 2\pi r
How do we apply this formula to find a physical distance on a circle?

Finding Arc Length

Let's find the arc length of a sector with a radius of r=9 cmr = 9\text{ cm} and a central angle of θ=80∘\theta = 80^\circ.