Geometry Lab reference
Arc length & sector area
Measure a slice of a circle.
s = rθ · A = ½r²θ
The idea
A central angle determines a fraction of a full turn. The same fraction applies to the circle’s circumference and its area. In radians, a full turn is 2π, which simplifies the formulas to arc length s = rθ and sector area A = r²θ/2.
Work through an example
For radius 5 and central angle 90°, θ = π/2. The arc is 5π/2 ≈ 7.854 units long and the sector area is 25π/4 ≈ 19.635 square units. The chord is shorter: 2 × 5 × sin 45° = 5√2 ≈ 7.071.
What to watch for
An arc follows the curved boundary; a chord is a straight segment. Sector area includes the region between the arc and two radii. Always convert degrees before using the radian formulas.
Make the formula move.
Explore this idea with real inputs.