5.1PureCore

Radian Measure

A radian is a more natural unit for measuring angles than the degree. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Radians make arc length, sector area, and calculus of trig functions far simpler.

25 min Radians

Edexcel A Level Maths: 5.1 Radians (Angles)Curated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Define a radian
Convert between degrees and radians
Know common angles in radians
Evaluate trig functions of radian angles

Degrees ↔ radians

A full turn is $360°$ or $2\pi$ radians, so $180° = \pi$ radians. To convert degrees to radians multiply by $\tfrac{\pi}{180}$ ; to convert radians to degrees multiply by $\tfrac{180}{\pi}$ .

180° = \pi \text{ rad}

The key conversion fact.

Common angles

Memorise: $30° = \tfrac{\pi}{6}$ , $45° = \tfrac{\pi}{4}$ , $60° = \tfrac{\pi}{3}$ , $90° = \tfrac{\pi}{2}$ , $360° = 2\pi$ .

Example 1Convert

135°

to radians.

135 \times \tfrac{\pi}{180} = \tfrac{135\pi}{180}

2Simplify:

\tfrac{3\pi}{4}

Answer

\tfrac{3\pi}{4}

radians

Tip — Keep answers as exact multiples of π unless told to use decimals.

Trig of radian angles

Trig functions take radian inputs directly: e.g. $\sin\tfrac{\pi}{6} = \tfrac12$ . In exams, set your calculator to radian mode when angles are in radians.

Formula recap

180° = \pi \text{ rad}

Base conversion.

\text{deg} \to \text{rad}: \times \tfrac{\pi}{180}

Degrees to radians.

\text{rad} \to \text{deg}: \times \tfrac{180}{\pi}

Radians to degrees.

Common mistakes to avoid

Leaving the calculator in degree mode for radian angles.

Switch to radian (RAD) mode when the angle is in radians.

Converting with the wrong factor.

Degrees→radians multiply by π/180; radians→degrees multiply by 180/π.

Key takeaways

180° = π radians is the key fact.
deg→rad ×π/180; rad→deg ×180/π.
Know π/6, π/4, π/3, π/2 by heart.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

$180°$ in radians is:
- $\pi$
- $2\pi$
- $\tfrac{\pi}{2}$
- $\tfrac{\pi}{180}$
Show answer and explanation
Answer: $\pi$
Definition.
$90°$ in radians is:
- $\tfrac{\pi}{2}$
- $\pi$
- $\tfrac{\pi}{4}$
- $2\pi$
Show answer and explanation
Answer: $\tfrac{\pi}{2}$
Half of π.
To convert degrees to radians, multiply by:
- $\tfrac{\pi}{180}$
- $\tfrac{180}{\pi}$
- $\pi$
- $180$
Show answer and explanation
Answer: $\tfrac{\pi}{180}$
deg→rad.
$135°$ in radians is:
- $\tfrac{3\pi}{4}$
- $\tfrac{2\pi}{3}$
- $\tfrac{\pi}{4}$
- $\tfrac{5\pi}{6}$
Show answer and explanation
Answer: $\tfrac{3\pi}{4}$
135π/180 = 3π/4.
$\sin\tfrac{\pi}{6}$ equals:
- $\tfrac12$
- $\tfrac{\sqrt3}{2}$
- $1$
- $\tfrac{\sqrt2}{2}$
Show answer and explanation
Answer: $\tfrac12$
π/6 = 30°.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

All 5 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.