1.4 Circular measure
- Syllabus
- 9709–2028–2029
- Topic
- 1.4
- Level
- AS
Learning objectives
1.4.1Radians• understand the definition of a radian, and use the relationship between radians and degrees1.4.2Arc length and sector area• use the formulae sr= i and Ar 2 1 2= i in solving problems concerning the arc length and sector area of a circle. Including calculation of lengths and angles in triangles and areas of triangles.
Radians make arc length and angle formulas proportional
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.
Sector area and arc length are fractions of a circle measured in radians
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.