IB Maths AI HLGeometry ToolkitPaper 1 & 2π = 180°~7 min read
Radian Measure
A radian is a different unit for angles — defined so that the arc length on a unit circle equals the angle. The whole circle measures 2π radians instead of 360°, so π radians = 180°. Converting between the two units is a single multiplication, and most exam questions in AI HL default to radians.
📘 What you need to know
Definition: 1 radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. On a unit circle, the arc length equals the angle in radians.
Master conversion: π rad = 180°. Everything else follows from this.
Notation: radians have no symbol by default (e.g., write “θ = π/3″, not “θ = π/3 rad”). The superscript c can be used but is rare.
Exam default: in AI HL Paper 1 & 2, work in radians unless the question explicitly uses degrees.
Why radians?
Radians measure angles by arc length on a unit circle. Imagine a circle of radius 1; mark a starting point on the circle and walk along the circumference. After walking a distance of 1 unit (one radius worth), you’ve swept an angle of exactly 1 radian. After walking 2 units, you’ve swept 2 radians. After walking around the entire circumference (2π units), you’ve swept 2π radians — a full turn. That’s why a full turn equals 2π rad and a half turn equals π rad. The relation π rad = 180° gives you everything else by proportion: divide both sides by anything to find smaller angles, multiply to find larger ones. 1 radian itself is approximately 57.3°, but you rarely need that decimal: most exam work stays in exact form using π.
The bold orange arc has length 1 (equal to the radius); the angle POQ at the centre is exactly 1 radian. The whole circumference of length 2π corresponds to a full angle 2π rad.
Always simplify fractions before computing. Converting 60° to radians: 60 × π/180 = 60π/180 = π/3 — cancel the 60 against the 180 first to avoid clunky fractions. Going the other way, converting 5π/4 to degrees: (5π/4) × (180/π) = 5 × 180/4 = 5 × 45 = 225° — the π cancels immediately. For non-standard angles like 43.8° or 2.5 rad, you get decimal answers; keep them to at least 4 d.p. during working and round at the end. In any “exact form” question, keep π throughout — don’t substitute π ≈ 3.14159.
The π trick for going to degrees: when converting kπ/n to degrees, just compute k × 180/n — the π on top and bottom cancels instantly. So 5π/6 = 5(180/6) = 5(30) = 150°.
🧭 Recipe — converting between radians and degrees
Identify the direction: are you going degrees → radians (multiply by π/180) or radians → degrees (multiply by 180/π)?
Simplify before multiplying: cancel common factors between the angle and 180.
For exact form, keep π in the answer; don’t approximate.
For decimals, work to at least 4 d.p. then round at the end.
Memorise the common angles: π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, π = 180°, 2π = 360° — these are worth instant recognition.
Worked examples
WE 1
Degrees → radians (exact form)
Convert each angle to radians in exact form: (a) 30°; (b) 90°; (c) 120°; (d) 270°.
multiply by 180/π2.5 × 180/π ≈ 2.5 × 57.296≈ 143.24°2.5 rad ≈ 143° (3 s.f.)just over 2 rad which is ≈ 114.6°; 2.5 rad is closer to a 4π/5 rotation (144°).
WE 5
Comparison: which angle is larger?
Which is larger, 100° or 2 radians? By how much, in degrees, to 3 s.f.?
convert both to the same unit100° in rad: 100π/180 = 5π/9 ≈ 1.745 radcompare to 2 rad2 rad > 5π/9 rad, so 2 rad is largerfind difference in degrees2 rad × 180/π ≈ 114.59°114.59 − 100 ≈ 14.59difference ≈ 14.6° (3 s.f.)
WE 6
Applied: angular speed of a wheel
A wheel completes 5 full revolutions plus an additional 240° in 12 seconds. (a) Find the total angle of rotation in degrees. (b) Express this angle in radians in exact form. (c) Find the average angular speed in rad/s, to 3 s.f.
