A radian is an angle measured by arc length: Wrap a piece of string equal to the radius r around the edge of a circle. The angle it spans at the centre is one radian — about 57.3°.
Going all the way round uses an arc of length 2πr, so a full turn is 2π radians.
That single fact is the whole conversion:
2π radians = 360°, so π radians = 180°.
IB-style question — degrees to radians
A wind-turbine blade turns through 150°.
Write this angle in radians, as an exact multiple of π.
Step by step
- To go from degrees to radians, multiply by π/180.
- Simplify the fraction 150/180 = 5/6.
Final answer
150° = 5π/6 radians (≈ 2.62 rad).
IB-style question — radians to degrees
A robot arm rotates by 0.8 radians.
Convert this to degrees, to one decimal place.
Step by step
- To go from radians to degrees, multiply by 180/π.
- Evaluate on the GDC.
Final answer
0.8 rad ≈ 45.8°.
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Why l = rθ is so clean: Arc length is the curved distance along the edge of the circle.
A full turn (θ = 2π) sweeps the whole circumference 2πr. A fraction of the turn sweeps that same fraction of the circumference — and because we used radians, the fraction is just θ/(2π).
So l = (θ/2π) × 2πr = rθ. The 2π cancels — but only because θ is in radians. Always switch to radians first.
IB-style question — length of a curved path
A cyclist rides round a circular roundabout of radius 12 m, turning through an angle of 1.4 radians at the centre.
Find the distance she travels along the curve.
Step by step
- Use arc length with the angle already in radians.
- Substitute r = 12, θ = 1.4.
Final answer
She travels 16.8 m along the curve.
IB-style question — convert first, then arc
A clock's minute hand is 9 cm long. Between 12:00 and 12:20 it sweeps 120°.
Find the distance the tip of the hand travels, to 3 significant figures.
Step by step
- Convert 120° to radians first (the formula needs radians).
- Apply l = rθ with r = 9.
- Evaluate.
Final answer
The tip travels 6π ≈ 18.8 cm. (If you forget to convert and use 120, you get the wrong answer.)