3 · IB Maths
Radians, arcs and sectors
One radian is the angle whose arc is as long as the radius. Open the angle and watch the arc length, sector area, chord and segment grow, with the angle in degrees or radians.
Predict first
Start from the default settings (r = 1.20 m, θ = 60°). Move the Angle θ slider to 120°. What happens to the chord length?
Chord = 2r sin(θ/2). Going from 60° to 120° changes sin 30° = 0.5 to sin 60° ≈ 0.866, a factor of about 1.73, while the arc length exactly doubles.
Check it in the model: Move the Angle θ slider from 60° to 120° and compare the Chord readout (1.20 m to 2.08 m).
arc s radius r chord and segment
s = rθ ; A = ½r²θ
chord = 2r sin(θ/2)
segment = ½r²(θ − sinθ)
Try this
- Choose the One radian preset and compare the radius bar with the arc bar. Then double the angle: what happens to s, A and the chord, and which of them does not double?
- Set r = 1.0 m and θ = 180°, then 360°. Check s = πr and s = 2πr, and explain why the chord is 2.00 m and 0 m.
- Keep θ = 60° and double the radius from 0.8 m to 1.6 m. By what factors do the arc length and sector area change?
Common error: The formulas s = rθ and A = ½r²θ only work when θ is in radians, so check the calculator mode and use π rad = 180°; the segment area is the sector minus the triangle, ½r²(θ − sinθ).
More on 3: Transformations of trigonometric graphs · Solving trigonometric equations · Volume and surface area of 3D solids · Ambiguous case of the sine rule · Vector equations of lines in 3D · Compound angle and double angle identities · Sine and cosine rule · The unit circle · Vectors and the dot product · Voronoi diagrams