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Step-by-Step Solution
Diagram
Sector
Arc (s)
Radius (r)
Chord (c)
Deriving Arc Length
An arc is a fraction of the full circumference. A central angle θ (in radians) spans a fraction of a full rotation (2π radians), so we scale the full circumference proportionally.
s = (θ / 2π) · 2πr
s = θr
s = rθ ✓
s = θr
s = rθ ✓
Deriving Sector Area
A sector is a "pie slice" — a fraction of the full circle. The full circle has area πr², so we scale by the fraction of the full angle (2π) that θ represents.
A = (θ / 2π) · πr²
A = θr² / 2
A = ½r²θ ✓
A = θr² / 2
A = ½r²θ ✓
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