Arc Length
Arc length is a fraction of the circumference: the actual distance along the curve, measured in linear units.
Introduction
Arc AB = fraction of circumference
Past Knowledge
Circumference (11.2.1). Arc measure in degrees (10.1.2).
Today's Goal
Calculate arc length using the proportion formula.
Future Success
Area of sectors (11.2.4), radian measure.
Key Concepts
Arc Length Formula
The arc is the same fraction of the circumference as the central angle is of 360°.
Key Distinction
Arc measure = degrees (how much of the circle). Arc length = linear distance (cm, ft, etc.).
Worked Examples
90° Arc
r = 10, θ = 90°.
ℓ = 5π ≈ 15.71
Find the Angle
Arc length = 12π, r = 18. Find θ.
→
θ = 120°
Common Pitfalls
Arc Length ≠ Arc Measure
A 90° arc in a tiny circle has a shorter length than a 90° arc in a huge circle. The degree measure is the same; the length is not.
Real-Life Applications
Running Track Curves
Each semicircular end of a track is an arc. The arc length determines staggered starting positions.
Pizza Crust
A pizza slice's outer crust is an arc. A 14" pizza cut into 8 slices: crust per slice = (1/8)(14π) ≈ 5.5 inches.
Practice Quiz
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