1.4 Circular measure

One radian is the angle subtended by an arc equal to the radius. A full turn is 2π radians, and θ radians correspond to arc length s=rθ.

Convert degrees before using calculus, arc length or sector formulas. Keep θ dimensionless in the formula and attach units to s or area.

A radius 5 cm and angle 1.2 rad give arc length 6.0 cm; the same numerical angle in degrees would give a wrong result.

A radian is not a physical length, and writing s=rθ with θ in degrees silently introduces a factor of π/180.

For radius r and angle θ in radians, arc length s=rθ and sector area A=½r²θ. Segment area requires subtracting the corresponding triangle from the sector.

Use the same angle units in every term, identify whether the question asks for sector or segment, and check that 0≤θ≤2π for an ordinary sector.

With r=4 and θ=π/3, s=4π/3 and sector area=8π/3. A minor segment would be this sector minus ½r²sinθ.

Sector area is not ½r²sinθ; that is the triangle area used when a segment is formed.