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Radian Measure

Q: How do I convert between degrees and radians?

Use = 180^. Degrees to radians: multiply by 180. Radians to degrees: multiply by 180. For example 90^ = 2.

A radian measures angle by arc length rather than by an arbitrary 360-degree convention. It looks unfamiliar at first, but it is the “natural” unit — and it makes arc length, sector area and later calculus dramatically simpler.

26 min Video by TLMaths Trigonometry

The big picture

Degrees are a human invention: 360 was chosen by the Babylonians, not by the mathematics. Radians instead define an angle by how far you travel around a unit circle, which ties angle directly to distance. That is why radians are compulsory for the arc and sector formulas here, and — more importantly — why every calculus result about trig functions (like the derivative of $sin x$ being $cos x$ ) only works in radians. Getting comfortable switching to radians now removes a whole category of future errors, starting with your calculator being in the right mode.

A-Level Maths E1-18: Using the Formula for Arc Length in RadiansWatch the walkthrough, then work through the notes below.

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What you'll be able to do

Define a radian and convert between degrees and radians
Know the standard angles in radians
Use $s = r θ$ for arc length
Use $A = \frac{1}{2} r^{2} θ$ for sector area

What a radian is

One $radian$ is the angle subtended at the centre of a circle by an arc equal in length to the radius. Wrap the radius along the circumference and the angle it spans is 1 radian. Since the full circumference is $2 π r$ , a full turn is $2 π$ radians — so $2 π$ radians $= 36 0^{\circ}$ , and $π$ radians $= 18 0^{\circ}$ .

That single equivalence, $π = 18 0^{\circ}$ , is the bridge for every conversion.

π radians = 18 0^{\circ}

Multiply degrees by

\frac{π}{180}

to get radians; multiply radians by

\frac{180}{π}

to get degrees.

6 0^{\circ} = 60 \times \frac{π}{180} = \frac{π}{3}

radians.

\frac{3 π}{4} \times \frac{180}{π} = \frac{3 ( 180 )}{4} = 13 5^{\circ}

Answer

6 0^{\circ} = \frac{π}{3}

;

\frac{3 π}{4} = 13 5^{\circ}

Tip — Learn the common conversions by heart: $3 0^{\circ} = \frac{π}{6}$ , $4 5^{\circ} = \frac{π}{4}$ , $6 0^{\circ} = \frac{π}{3}$ , $9 0^{\circ} = \frac{π}{2}$ .

Arc length and sector area

Radians make the arc and sector formulas beautifully simple — but only because the angle is in radians. The arc length is just radius times angle, and the sector area is half the radius squared times the angle.

If you ever plug a degree value into these, the answer is nonsense — so convert to radians first, every time.

s = r θ

Arc length (

θ

in radians).

A = \frac{1}{2} r^{2} θ

Sector area (

θ

in radians).

1Arc length:

s = r θ = 5 \times 0.8 = 4

cm.

2Area:

A = \frac{1}{2} r^{2} θ = \frac{1}{2} (25) (0.8) = 10

cm².

AnswerArc length 4 cm, area 10 cm².

Why this works

These formulas are why radians exist: “arc = radius × angle” only works because a radian is $defined$ by arc length. In degrees you would need clumsy factors of $\frac{π}{180}$ everywhere.

Radians everywhere from here

From this point on, exact-value questions are usually posed in radians ( $sin \frac{π}{6} = \frac{1}{2}$ ), and all of calculus with trig functions assumes radians. The main practical consequence is your $calculator mode$ : switch it to radians whenever the angle is given in $π$ terms, and back to degrees when it is not.

Think like an examiner

The arc and sector formulas require the angle in $radians$ — convert first if a degree value is given.
Match your calculator mode to the question; a mode mismatch is one of the most common silent errors.
Leave answers in terms of $π$ when the question says “exact” — do not convert $\frac{π}{3}$ to a decimal.
Learn the standard angle conversions so you can move between the two systems instantly.

Common misconceptions

Using

s = r θ

with

θ

in degrees.

Convert to radians first; the formula only holds for radian angles.

Leaving the calculator in degree mode for a radian question.

Switch to radian mode whenever the angle is in

π

terms.

Writing

π = 36 0^{\circ}

π

radians

= 18 0^{\circ}

; a full turn

2 π = 36 0^{\circ}

Rounding

\frac{π}{6}

to a decimal in an exact-answer question.

Keep it as

\frac{π}{6}

(or the exact surd value it gives) when exact form is required.

Radian essentials

π = 18 0^{\circ}

Degree–radian bridge.

s = r θ

Arc length.

A = \frac{1}{2} r^{2} θ

Sector area.

\frac{π}{6}, \frac{π}{4}, \frac{π}{3}, \frac{π}{2}

Standard angles.

Stretch yourself

A sector of a circle has radius 6 cm and perimeter 20 cm. Find the angle of the sector in radians, and its area.

Hint — The perimeter is two radii plus the arc: $2 r + s = 20$ . Find the arc length $s$ , then use $s = r θ$ and $A = \frac{1}{2} r^{2} θ$ .

Questions students ask

What exactly is a radian?

The angle at the centre of a circle subtended by an arc equal in length to the radius. Because the circumference is

2 π r

, a full turn is

2 π

radians, so

π

radians equals

18 0^{\circ}

How do I convert between degrees and radians?

Use

π = 18 0^{\circ}

. Degrees to radians: multiply by

\frac{π}{180}

. Radians to degrees: multiply by

\frac{180}{π}

. For example

9 0^{\circ} = \frac{π}{2}

Why must the angle be in radians for

s = r θ

Because a radian is defined so that arc length equals radius times angle. In degrees you would need an extra factor of

\frac{π}{180}

, so the clean formula only works in radians.

Why do we bother with radians at all?

They are the natural unit that links angle to distance, which is why the arc and sector formulas are simple and why calculus of trig functions (e.g.

\frac{d}{d x} sin x = cos x

) only holds in radians.

Key takeaways

A radian is the angle whose arc equals the radius; $π$ radians $= 18 0^{\circ}$ .
Convert with $\frac{π}{180}$ (deg→rad) and $\frac{180}{π}$ (rad→deg).
Arc length $s = r θ$ ; sector area $A = \frac{1}{2} r^{2} θ$ — angle in radians.
Match your calculator mode to the question.
Radians are essential for later calculus with trig functions.

How this fits the course

Build on

Trigonometric Ratios and Graphs

The Sine and Cosine Rules

Leads to

Trigonometric Identities

Test yourself

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