01 · Meaning
An arc has length 12 cm on a circle of radius 8 cm. Find its central angle in radians.
Hint
Divide arc length by radius.
Worked solution
θ = 12/8 = 3/2 rad = 1.5 rad.
Understand · explore · practise
Radians and degrees
Understand one radian, convert degrees and radians exactly, and interpret signed angles and full turns. Move the circle yourself, then practise with worked solutions.
Before you startCircle radius and circumference; fractions and π
01 / What is one radian?
Measure a turn using the circle itself.
Take a circle and mark an arc whose length is exactly one radius. The angle that arc makes at the centre is one radian. A radian is an angle unit, just as a degree is, but it comes directly from circle geometry.
Angle in radians = arc length ÷ radius
Use the same length unit for the arc and radius. The ratio has no length unit.
The arc is the curved part of the circumference. It is not the straight chord joining its endpoints. The definition needs the angle at the centre, not an angle elsewhere on the circle.
Keep the angle; change the circleExplore
Blue lines: radii. Gold curve: the chosen anticlockwise arc. All lengths use the same unit.
Arc ÷ radius = π ÷ 2 = π/2 rad ≈ 1.570796 rad.
Change the radius while leaving the angle fixed. The arc changes by the same scale factor, so their ratio stays the same.
Watch the radius and arc scale together
Pause, replay or seek freely. The notes explain the same idea and stay in view.
02 / Why the radius cancels
A bigger circle does not make the same turn a bigger angle.
Two circles, the same central angleWorked example
Circle A: radius 4 cm, arc length 6 cm
The radian measure is 6/4 = 3/2.
Circle B: radius 10 cm, arc length 15 cm
The ratio is 15/10 = 3/2 again.
Both angles are 1.5 radians
Every length was multiplied by 2.5, so the ratio stayed fixed.
If an arc is 35 mm long on a circle of radius 5 cm, first convert 5 cm to 50 mm. The angle is 35/50 = 0.7 rad, not 7 rad. A ratio of lengths is meaningful only after using a common unit.
03 / A full turn is 2π radians
The circumference supplies the conversion.
Full turn: θ = 2πr/r = 2π rad = 360°
Half turn: π rad = 180°
Therefore 1 rad = 180/π degrees ≈ 57.3°. One radian is not π degrees, and one full turn is not one radian.
Useful fractions of a turnWorked example
Quarter turn: π/2 rad = 90°
Divide a full turn by four.
One sixth of a turn: π/3 rad = 60°
Divide 2π by six.
One eighth of a turn: π/4 rad = 45°
Divide 2π by eight.
One twelfth of a turn: π/6 rad = 30°
Divide 2π by twelve.
π is a number, approximately 3.14159. The expression π rad is an exact angle; a decimal such as 3.14 rad is usually only an approximation to it.
04 / Degrees to radians
Multiply by π/180 and simplify the fraction.
θ radians = angle in degrees × π/180
Convert 225° and −72°Worked example
225° = (225/180)π rad = 5π/4 rad
Divide numerator and denominator by 45.
−72° = −(72/180)π rad = −2π/5 rad
The negative sign records the direction of rotation.
Keep π in an exact answer unless a decimal is requested. For 225°, 3.92699… is less useful than 5π/4 when recognising exact trig values later.
05 / Radians to degrees
Multiply by 180/π.
Angle in degrees = θ radians × 180/π
Convert an exact multiple of π and a decimalWorked example
7π/12 rad = (7π/12)(180/π)° = 105°
The π factors cancel.
2.3 rad = (414/π)° ≈ 131.780…°
Here there is no π in the given numerator to cancel.
2.3 rad ≈ 132° to 3 significant figures
Round only the final result.
Quick check: π rad is 180°, so a positive angle smaller than π rad must convert to less than 180°. This helps catch the conversion factor being upside down.
06 / Signed angles and extra turns
The final direction does not tell you how far you travelled.
Starting from the positive horizontal ray, the usual convention is positive for anticlockwise rotation and negative for clockwise rotation. An angle can exceed one full turn.
Same final ray, different rotationsWorked example
13π/6 = 2π + π/6
This is one full anticlockwise turn, then another 30°.
−11π/6 + 2π = π/6
A clockwise 330° rotation ends on the same ray as 30°.
All coterminal angles: π/6 + 2kπ, k an integer
Adding whole turns preserves the endpoint direction.
These are distinct angles with the same terminal ray. If asked for the representative in 0 ≤ θ < 2π, use π/6. The upper endpoint 2π is excluded, so a full turn has representative 0.
07 / Rotation is signed; distance is nonnegative
Keep the meaning of the angle clear.
For a nonnegative central angle θ, the corresponding arc length is rθ. If a point rotates clockwise through a negative angle, its distance travelled is r|θ|. A signed rotation and a physical length are different quantities.
A point moves clockwise through −5π/2 on radius 3 cmWorked example
Magnitude of the turn = 5π/2 rad
This is one and a quarter turns.
Distance travelled = 3 × 5π/2 = 15π/2 cm
Use the whole rotation, not only its final direction.
Terminal ray = 3π/2 in 0 ≤ θ < 2π
Add 4π to −5π/2. Reducing the angle changes the described journey, so do not use it to find the distance travelled.
When solving an ordinary sector problem, choose the specified minor or major angle. Do not silently replace a reflex angle with its smaller complement.
08 / Calculator mode and notation
The same typed number can mean two different angles.
Check your angle modeWorked example
In RAD mode: sin(π/6) = 1/2
The input is a radian measure.
In DEG mode: sin(30) = 1/2
The input is a degree measure.
In RAD mode: sin(30) is not 1/2
30 radians is several full turns, not 30 degrees.
Use RAD mode for radian trig inputs and inverse-trig outputs. Ordinary arithmetic such as 225 × π/180 does not depend on angle mode. If an angle is explicitly written with °, it is in degrees. In these Pure 2 lessons, a trig input without ° is in radians unless stated otherwise.
A bare number can be a radian measure: 1.2 rad is perfectly valid. Radians do not have to contain π.
09 / Your turn
Keep exact multiples of π; round only when asked.
02 · Units first
An arc is 42 mm long and the radius is 3 cm. Find θ.
Hint
Convert the radius to millimetres.
Worked solution
3 cm = 30 mm. θ =42/30 =7/5 rad.
03 · To radians
Convert 150° and 315° to radians.
Hint
Multiply each by π/180.
Worked solution
150π/180 =5π/6 rad;315π/180 =7π/4 rad.
04 · A small exact angle
Convert 22.5° to radians.
Hint
22.5/180 =1/8.
Worked solution
π/8 rad. A decimal degree input can still give an exact fractional multiple ofπ.
05 · Negative and large
Convert −210° and 810° to radians without reducing the rotations.
Hint
Keep the signs and all full turns.
Worked solution
−210π/180 =−7π/6 rad;810π/180 =9π/2 rad.
06 · To degrees
Convert 11π/12 rad and −5π/3 rad to degrees.
Hint
Multiply by 180/π.
Worked solution
165° and−300°, respectively.
07 · Decimal radians
Convert 0.8 rad to degrees to 3 significant figures.
Hint
The exact degree value is 144/π.
Worked solution
144/π ≈45.8366…°, so 45.8°.
08 · Same terminal ray
Find the representatives of 17π/6 and −7π/4 in 0 ≤ θ < 2π.
Hint
Subtract or add whole multiples of 2π.
Worked solution
17π/6 −12π/6 =5π/6;−7π/4 +8π/4 =π/4.
09 · A full turn boundary
What is the representative of 6π in 0 ≤ θ < 2π? How many turns does 6π describe?
Hint
Distinguish the final ray from the total rotation.
Worked solution
The representative is 0. The original rotation is 3 full anticlockwise turns.
10 · Scale the circle
On radius 5, an arc subtends 1.2 rad. Another circle has radius 15 and the same angle. Find both arc lengths.
Hint
Rearrange θ=l/r as l=rθ.
Worked solution
Lengths 6 and 18 in the same units as their radii. The factor 3 changes both radius and arc, leaving the ratio 1.2 unchanged.
11 · Clockwise travel
A point on radius 4 m rotates through −3π/2. Find the distance travelled and the terminal-ray representative in 0 ≤ θ < 2π.
Hint
Use the angle magnitude for distance; add 2π for the representative.
Worked solution
Distance 4 ×3π/2 =6π m. Representativeπ/2 rad.
12 · Diagnose the error
A student says “one radian is 180°, since π radians is 180°”. Correct this and explain whether a larger circle changes the answer.
Hint
Divide both sides ofπ rad = 180° by π.
Worked solution
One radian is 180/π degrees, about 57.3°. A larger circle scales arc and radius equally; it does not change this conversion.
10 / Recap
A radian compares two lengths; conversions compare two angle units.
- One radian subtends an arc equal to the radius.
- θ=l/r requires matching length units.
- 2π radians make one full turn.
- Multiply degrees by π/180; multiply radians by 180/π.
- Keep exact multiples ofπ when possible.
- Whole turns preserve the terminal ray but still count towards travel.
- Check RAD mode for trig calculations in radians.
Section 1 of 10 · What is one radian?