MathsISC Class 11 · Trigonometric Functions

Radian Measure, Arc Length and Sector Area: Conversions and Applications

Updated 17 October 2026 · Fact-checked

One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. π radians = 180°, so 1° = π/180 rad and 1 rad = 180°/π ≈ 57.3°. For a circle of radius r and central angle θ (in radians) the arc length is s = rθ and the area of the sector is ½r²θ = ½rs.

Understand Radian Measure, Arc Length and Sector Area

Measures of angles. In the sexagesimal system, 1° = 60′ and 1′ = 60″. In the radian system an angle θ = s/r, where s is the arc length. Since the full circle has arc $2 π r$ , one revolution is $2 π$ radians, so $π$ rad = 180°.

Conversions. Degrees to radians: multiply by $π /180$ . Radians to degrees: multiply by $180/ π$ . Always convert minutes and seconds to decimal degrees first: 40°20′ = 40 + 20/60 degrees. If the question says take π = 22/7, use it consistently.

Arc length and sector. For central angle θ in radians: $s = r θ$ and area $A = \frac{1}{2} r^{2} θ$ . If θ is in degrees, first convert it to radians. The perimeter of a sector is $2 r + r θ$ . A larger angle with the same radius gives a longer arc and a larger area in direct proportion.

Rotating bodies. If a wheel of radius r makes n revolutions per second, the angle turned per second is $2 π n$ rad (the angular speed) and the linear speed of the rim is $2 π n r$ . The tip of a clock hand turns through $2 π$ rad in one hour for the minute hand and in 12 hours for the hour hand.

Checking. A quick test: a right angle is $π /2 \approx 1.571$ rad; a straight angle is 3.1416 rad; one radian is about 57°17′45″.

Radian Measure, Arc Length and Sector Area: formulas to remember

Conversion: $180° = \pi$ rad | $1° = \frac{\pi}{180}$ rad | $1$ rad $= \frac{180°}{\pi}$; Convert minutes to degrees first.
Arc length: $$s = r θ$$; θ in radians.
Sector area: $A = \frac{1}{2} r^2 θ = \frac{1}{2} r s$; θ in radians.
Rotation: $angular speed $ω = 2\pi n$ | linear speed $v = r ω$$; n = revolutions per unit time.

Radian Measure, Arc Length and Sector Area in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Distance rolled by a wheel

Suppose a bicycle wheel of radius 35 cm turns 3 times every second.

Find the angle turned per second, the speed of a point on the rim and the distance travelled by the bicycle in 1 minute.

Try it first, then show the solutionSolution

Angle per second $= 3 \times 2 π = 6 π$ rad.
Rim speed $= r ω = 0.35 \times 6 π = 2.1 π = 6.60$ m s⁻¹.
Distance in 60 s $= 6.60 \times 60 = 395.8$ m (that is, 180 revolutions of circumference $2 π \times 0.35 = 2.199$ m).

Answer: 6π rad s⁻¹; 6.60 m s⁻¹; about 396 m.

Memory hook: Distance rolled = circumference × number of turns.

Tricks to remember Radian Measure, Arc Length and Sector Area

Short memory aids. Say each one aloud once, then use it on a question the same day.

π is 180
Replace π by 180° to switch between radians and degrees.
Use it when: Quick conversions.
s = rθ only in radians
If the angle is given in degrees, convert first.
Use it when: Arc length questions.

Radian Measure, Arc Length and Sector Area: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Degrees to radians · 1 mark · MCQBoard level

75° expressed in radians is:
(a) $\frac{5 π}{12}$
(b) $\frac{7 π}{12}$
(c) $\frac{5 π}{6}$
(d) $\frac{3 π}{8}$

Try it first, then show the solutionSolution

$75 \times \frac{π}{180} = \frac{5 π}{12}$ .

Answer: (a) $\frac{5 π}{12}$

Example 2: Conversions · 3 marks · Short answerBoard level

Express 40°20′ in radians and $\frac{5 π}{9}$ radians in degrees.

Try it first, then show the solutionSolution

$40°20' = 40 + \frac{20}{60} = \frac{121}{3}$ degrees. (1 mark)
In radians: $\frac{121}{3} \times \frac{π}{180} = \frac{121 π}{540} \approx 0.704$ . (1 mark)
$\frac{5 π}{9} \times \frac{180}{π} = 100°$ . (1 mark)

Answer: $\frac{121 π}{540}$ rad ≈ 0.704 rad; 100°.

Example 3: Pendulum arc and sector · 4 marks · Short answerHarder board level

The bob of a pendulum of length 50 cm swings through an angle of 72°. Find the length of the arc described and the area of the sector swept.

Try it first, then show the solutionSolution

$72° = 72 \times \frac{π}{180} = \frac{2 π}{5}$ rad. (1 mark)
Arc length $s = r θ = 50 \times \frac{2 π}{5} = 20 π \approx 62.8$ cm. (1.5 marks)
Area $= \frac{1}{2} r^{2} θ = \frac{1}{2} \times 2500 \times \frac{2 π}{5} = 500 π \approx 1570.8$ cm². (1.5 marks)

Answer: Arc 20π ≈ 62.8 cm; area 500π ≈ 1571 cm².

Example 4: Angle from an arc · 4 marks · Short answerHarder board level

An arc of length 22 cm subtends an angle at the centre of a circle of radius 14 cm. Find the angle in radians and in degrees (take π = 22/7). Find also the perimeter and area of the sector.

Try it first, then show the solutionSolution

$θ = s / r = 22/14 = \frac{11}{7}$ rad. (1 mark)
In degrees: $\frac{11}{7} \times \frac{180}{22/7} = 90°$ . (1 mark)
Perimeter $= 2 r + s = 28 + 22 = 50$ cm. (1 mark)
Area $= \frac{1}{2} r s = \frac{1}{2} \times 14 \times 22 = 154$ cm². (1 mark)

Answer: 11/7 rad = 90°; perimeter 50 cm; area 154 cm².

Example 5: Minute hand of a clock · 5 marks · Long answerHOTS / competency

The minute hand of a clock is 7 cm long. Find the angle turned, the distance covered by its tip and the area swept in 40 minutes.

Try it first, then show the solutionSolution

In 60 min the hand turns $2 π$ ; in 40 min $θ = \frac{40}{60} \times 2 π = \frac{4 π}{3}$ rad. (1.5 marks)
Distance by tip $= r θ = 7 \times \frac{4 π}{3} = \frac{28 π}{3} \approx 29.3$ cm. (1.5 marks)
Area swept $= \frac{1}{2} r^{2} θ = \frac{1}{2} \times 49 \times \frac{4 π}{3} = \frac{98 π}{3} \approx 102.6$ cm². (1.5 marks)
Check: the area equals ½ × r × s = ½ × 7 × 29.3 = 102.6 cm². (0.5 mark)

Answer: 4π/3 rad; ≈ 29.3 cm; ≈ 102.6 cm².

Where students lose marks in Radian Measure, Arc Length and Sector Area

The slips examiners see most often, and what to write instead.

Mistake: Using degrees in s = rθ
Putting θ = 72 in s = rθ.
Do this: Convert to radians before using s = rθ.
Usually costs: 2 marks.
Mistake: Dropping the minutes
Converting 40°20′ as 40°.
Do this: Convert 20′ to 1/3 of a degree.
Usually costs: 1 mark.
Mistake: Mixing the sector area formula
Using r²θ.
Do this: Area is ½r²θ.
Usually costs: 1 to 2 marks.

Exam tips

Write π rad = 180° as the first line of every conversion.
Keep answers in terms of π unless a decimal is requested.
State the units: cm for arc, cm² for area.
Check with ½ r s for the sector area.

Practice questions on Radian Measure, Arc Length and Sector Area

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The radian measure of 210° is:
(a) $\frac{5 π}{6}$
(b) $\frac{7 π}{6}$
(c) $\frac{4 π}{3}$
(d) $\frac{3 π}{2}$
Hint
Multiply by π/180.
Answer
(b) $\frac{7 π}{6}$
Q2 · 2 marks · Short answerBoard level
Convert $\frac{7 π}{12}$ radians into degrees.
Hint
Multiply by 180/π.
Answer
$\frac{7 π}{12} \times \frac{180}{π} = 105°$ .
Q3 · 3 marks · Short answerHarder board level
The three angles of a triangular plot are in the ratio 2 : 3 : 7. Find each angle in radians and in degrees.
Hint
The sum is π.
Answer
Parts total 12: $\frac{2 π}{12} = \frac{π}{6}$ (30°), $\frac{3 π}{12} = \frac{π}{4}$ (45°), $\frac{7 π}{12}$ (105°).
Q4 · 3 marks · Short answerHarder board level
A wheel of radius 0.7 m makes 5 revolutions per second (π = 22/7). Find the angle turned in one minute and the distance covered by the rim in one minute.
Hint
Revolutions in a minute: 300.
Answer
Angle $= 300 \times 2 π = 600 π$ rad. Distance $= 300 \times 2 π r = 300 \times 2 \times 22/7 \times 0.7 = 300 \times 4.4 = 1320$ m.
Q5 · 3 marks · Short answerHarder board level
A circular arc of length 15 cm subtends 60° at the centre. Find the radius and the area of the sector.
Hint
Convert 60° to π/3.
Answer
$r = \frac{s}{θ} = \frac{15}{π /3} = \frac{45}{π} \approx 14.32$ cm; area $= \frac{1}{2} r s = \frac{1}{2} \times 14.32 \times 15 \approx 107.4$ cm².
Q6 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a sprinkler on a lawn waters a sector of radius 8 m through an angle of 135°.
(i) Find the angle in radians. (1 mark)
(ii) Find the length of the boundary arc. (1 mark)
(iii) Find the area watered. (1 mark)
(iv) Find the angle needed to water an area of 50 m². (1 mark)
Hint
Area = ½r²θ.
Answer
(i) $135° = \frac{3 π}{4}$ rad. (ii) $s = 8 \times \frac{3 π}{4} = 6 π \approx 18.85$ m. (iii) $A = \frac{1}{2} \times 64 \times \frac{3 π}{4} = 24 π \approx 75.4$ m². (iv) $θ = \frac{2 A}{r ^{2}} = \frac{100}{64} = 1.5625$ rad ≈ 89.5°.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.1 radian in degrees?Show answer
About 57.3°.
2.Arc length?Show answer
rθ.
3.Sector area?Show answer
½ r² θ.
4.π radians equals?Show answer
180°.

Radian Measure, Arc Length and Sector Area: frequently asked questions

Why do mathematicians prefer radians?

Because s = rθ and the derivatives of sine and cosine take their simplest forms only when angles are measured in radians.

Is the radian a unit of length?

No. It is the ratio of two lengths (arc/radius), so it is dimensionless.

Can θ exceed 2π?

Yes. For rotations through more than one full turn, θ is larger than 2π.