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Learn7357Paper 2 · Pure and MechanicsPure content (sections A–J)Radians
Pure content (sections A–J) · Subtopic 10 of 66

Radians

What a radian is, why mathematicians prefer it to the degree, and how to convert between the two, with exact values in terms of π\pi.

Cambridge 9709 Paper 1Edexcel YMA01 WMA11Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2AQA 7357 Paper 1AQA 7357 Paper 2≈ 30 minRevision 6 · updated 2026-10-27
Section 1 of 14

Learning objectives

  • Explain what one radian is, using the arc and radius of a circle.
  • Convert between degrees and radians, exactly in terms of π\pi and as decimals.
  • Recognise the common angles π6, π4, π3, π2\tfrac{\pi}{6},\ \tfrac{\pi}{4},\ \tfrac{\pi}{3},\ \tfrac{\pi}{2} and their trigonometric values.
  • Use radians to describe turning, such as a wheel or a clock hand.
Pause and think

Take a piece of string as long as the radius of a circle and lay it around the edge. How many of those strings fit around the whole circumference?

Section 2 of 14

Core idea

Start hereWhy 360?

Why does a full turn have 360∘360^\circ? Probably because ancient Babylonians liked numbers that divide nicely, and a year has roughly 360360 days. It's a human choice. Is there a natural way to measure angles, one that comes from the circle itself? Yes: wrap the radius around the edge. The angle it covers is one radian, and it makes all of later maths (arcs, areas, calculus of sin⁡x\sin x) far simpler.

The big idea

One radian is the angle at the centre when the arc length equals the radius. A full turn is 2π2\pi radians, so π rad=180∘\pi\text{ rad}=180^\circ. Convert with ×π180\times\dfrac{\pi}{180} (degrees → radians) or ×180π\times\dfrac{180}{\pi} (radians → degrees).

1. Defining the radian

One radian: arc length equals the radius
rr1 radarc = r
Bend the radius around the circle: the angle it covers is 11 radian, about 57.3∘57.3^\circ. The circumference is 2πr2\pi r, so a full turn is 2π2\pi radians.
How many radians in a full turn?
  1. circumference=2πr\text{circumference}=2\pi r
    Known formula.
  2. number of radii that fit=2πrr=2π\text{number of radii that fit}=\frac{2\pi r}{r}=2\pi
    Each radius-length of arc is one radian.
  3. 2π rad=360∘ ⇒ π rad=180∘2\pi\text{ rad}=360^\circ\ \Rightarrow\ \pi\text{ rad}=180^\circ
    Halve.
  4. 1 rad=180∘π≈57.3∘1\text{ rad}=\frac{180^\circ}\pi\approx57.3^\circ
    Divide by π\pi.
Sweeping round the circle: degrees and radians together
half turn = π ≈ 3.14 · full turn = 2π ≈ 6.28
Watch the value: a quarter turn is π2≈1.57\tfrac{\pi}{2}\approx1.57, a half turn π≈3.14\pi\approx3.14, a full turn 2π≈6.282\pi\approx6.28.

2. Converting

Worked exampleConversions

Convert 60∘60^\circ and 135∘135^\circ to radians, and 5π6\frac{5\pi}6 and 2.52.5 rad to degrees.

Goal
Exact radians in terms of π\pi; degrees.
Strategy
Multiply by π180\frac\pi{180} or 180π\frac{180}\pi.
  1. 60×π180=π3,135×π180=3π460\times\frac\pi{180}=\frac\pi3,\qquad135\times\frac\pi{180}=\frac{3\pi}4
    Simplify the fractions.
  2. 5π6×180π=150∘\frac{5\pi}6\times\frac{180}\pi=150^\circ
    The π\pi's cancel.
  3. 2.5×180π=143.2∘2.5\times\frac{180}\pi=143.2^\circ
    Calculator.
Answerπ3\frac\pi3, 3π4\frac{3\pi}4, 150∘150^\circ, 143.2∘143.2^\circ.
✓ Checkπ3≈1.047\frac\pi3\approx1.047 rad ≈1.047×57.3≈60∘\approx1.047\times57.3\approx60^\circ ✓.
  • 30∘=π630^\circ=\frac\pi6, 45∘=π445^\circ=\frac\pi4, 60∘=π360^\circ=\frac\pi3, 90∘=π290^\circ=\frac\pi2, 180∘=π180^\circ=\pi, 270∘=3π2270^\circ=\frac{3\pi}2, 360∘=2π360^\circ=2\pi.
Common misconceptionCalculator in the wrong mode sin⁡π6\sin\frac\pi6 should be 0.50.5. If you get 0.009140.00914, the calculator is in degrees. Switch to radians (R or RAD).
Edge caseUnit symbol An angle written without a degree sign, like θ=2\theta=2, is in radians by convention.
Pause and think

Convert 7π4\frac{7\pi}{4} to degrees and 20∘20^\circ to radians.

Summary

  • 1 rad: arc = radius.
  • π\pi rad =180∘=180^\circ.
  • Use radians mode.
Key words
Radian
The angle whose arc equals the radius.
HistoryWhy 360, and who named the radian? The 360 degrees in a turn probably come from Babylonian astronomers, who counted in base 60 and noticed a year is roughly 360 days. The idea of the radian appears in the work of Roger Cotes around 1714, but the name "radian" was first printed in 1873, in exam questions set by the engineer James Thomson, the brother of Lord Kelvin.
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