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CBSE · Class 11 · Mathematics

Angles and Radian Measure (Ch 3: 3.2)

4 concepts · 32 practice questions with answers · Free to read

Angle and degree measure

From daily life

The minute hand turns a full circle in an hour, and a ceiling fan keeps spinning in one direction. An angle is the amount of turning.

Simple definition

An angle is the amount of rotation of a ray from its initial side to its terminal side. Anticlockwise rotation gives a positive angle, clockwise a negative angle. One full turn = 360°, so 1° = 1/360 of a turn, 1° = 60′ (minutes) and 1′ = 60″ (seconds).

xy135°

In the figure: The angle 135° is measured anticlockwise from the positive x-axis; it ends in the second quadrant.

Rule 1: Quadrant of an angle — 0°–90° is I, 90°–180° is II, 180°–270° is III, 270°–360° is IV.

  • Easy: 40° lies in quadrant I.
  • Medium: 210° lies in quadrant III.
  • Exam-type: −60° lies in quadrant IV (clockwise 60° equals 300°).

Rule 2: Degrees, minutes, seconds — Convert fractions of a degree by multiplying by 60.

  • Easy: 0.5° = 30′.
  • Medium: 25.25° = 25° 15′ (0.25 × 60 = 15′).
  • Exam-type: 1 hour of the clock hand = 30°; so 20 minutes of the minute hand = 120°.
Remember

Angles that differ by a full turn (360°) end on the same terminal side, e.g. 30° and 390° and −330°.

Common mistake

Treating clockwise rotation as positive.

Memory clue

Anticlockwise = Added (positive).

Practice questions

Easy
  1. 1In which quadrant does 100° lie?
  2. 2Convert 0.75° to minutes.
  3. 3Name an angle coterminal with 50°.
Medium
  1. 4In which quadrant does −150° lie?
  2. 5Convert 12° 30′ to degrees.
  3. 6Through what angle does the hour hand turn in 2 hours?
Challenging
  1. 7Through what angle does the minute hand turn in 25 minutes?
  2. 8A wheel turns 2 full turns clockwise. State the angle with its sign.
Show answers (try them yourself first)
  1. 1Quadrant II
  2. 245′
  3. 3410° (or −310°)
  4. 4−150° = 210°, quadrant III
  5. 512.5°
  6. 660°
  7. 7150°
  8. 8−720°

Radian measure

From daily life

To measure length, you pick a unit like a metre. To measure turning in a neat way for the circle itself, we use the circle's own radius.

Simple definition

One radian is the angle at the centre of a circle made by an arc whose length equals the radius. For an arc of length l on a circle of radius r, the angle is θ = l / r radians. A full circle (circumference 2πr) is 2π radians.

Formula: θ = l / r

Rule 1: Find θ in radians — Divide arc by radius (both in the same unit).

  • Easy: l = 10 cm, r = 5 cm: θ = 2 radians.
  • Medium: l = 6 cm, r = 4 cm: θ = 1.5 radians.
  • Exam-type: l = π cm, r = 2 cm: θ = π/2 radians.

Rule 2: Find the arc length — Rearrange to l = rθ (θ in radians).

  • Easy: r = 7 cm, θ = 2 radians: l = 14 cm.
  • Medium: r = 10 cm, θ = π/5: l = 2π cm ≈ 6.28 cm.
  • Exam-type: r = 6 m, θ = 0.5 radian: l = 3 m.
Remember

Radian is a ratio (arc ÷ radius), so it has no unit; we may write rad or just leave it out.

Common mistake

Using θ = l/r when θ is given in degrees. First convert to radians.

Memory clue

Radian: Ratio of Arc to Radius.

Practice questions

Easy
  1. 1l = 12 cm, r = 3 cm. Find θ.
  2. 2r = 5 cm, θ = 3 radians. Find l.
  3. 3l = 9 cm, θ = 1.5 radians. Find r.
Medium
  1. 4r = 14 cm, θ = π/7. Find l using π = 22/7.
  2. 5A circle has r = 10 cm. An arc of 25 cm subtends what angle?
  3. 6How many radians in one full circle?
Challenging
  1. 7The minute hand of a clock is 7 cm long. How far does its tip move in 15 minutes? (π = 22/7)
  2. 8Two arcs of the same circle have lengths 8 cm and 12 cm. Find the ratio of their angles.
Show answers (try them yourself first)
  1. 14 radians
  2. 215 cm
  3. 36 cm
  4. 4l = 14 × π/7 = 2π ≈ 6.28 cm
  5. 52.5 radians
  6. 62π
  7. 7Angle = π/2, l = 7 × π/2 = 11 cm
  8. 88 : 12 = 2 : 3

Degrees and radians

From daily life

Kilometres and miles measure the same distance in different units. Degrees and radians measure the same angle.

Simple definition

Since 360° = 2π radians, 180° = π radians. So 1° = π/180 radian and 1 radian = 180°/π ≈ 57°17′45″.

Formula: Radian = Degree × π/180 Degree = Radian × 180/π

Rule 1: Degree to radian — Multiply by π/180.

  • Easy: 60° = π/3.
  • Medium: 150° = 5π/6.
  • Exam-type: −225° = −5π/4.

Rule 2: Radian to degree — Multiply by 180/π.

  • Easy: π/6 = 30°.
  • Medium: 7π/4 = 315°.
  • Exam-type: 2 radians = 114.6° (using the exact ratio).
Remember

Values to learn: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π.

Common mistake

Writing 180° = 3.14 instead of π radians.

Shortcut

Replace π by 180° to switch quickly: 2π/3 = 2 × 180°/3 = 120°.

Memory clue

π means a half-turn, 180°.

Spot the difference

DegreeRadian
Full turn 360°Full turn 2π
Handy for everydayHandy for formulas like l = rθ

Practice questions

Easy
  1. 1Convert 120° to radians.
  2. 2Convert 3π/4 to degrees.
  3. 3Convert −90° to radians.
Medium
  1. 4Convert 5π/12 to degrees.
  2. 5Convert 40° to radians.
  3. 6Find the angle in radians turned by a wheel making 3 full turns.
Challenging
  1. 7The angles of a triangle are in the ratio 2 : 3 : 4. Find the largest angle in radians.
  2. 8In a regular hexagon each interior angle is 120°. Write it in radians and the exterior angle in degrees.
Show answers (try them yourself first)
  1. 12π/3
  2. 2135°
  3. 3−π/2
  4. 475°
  5. 52π/9
  6. 66π
  7. 7180° × 4/9 = 80° = 4π/9
  8. 82π/3 and 60°

Arc length and sector

From daily life

The curved edge of a pizza slice is an arc; its length depends on the slice angle and the pizza radius.

Simple definition

On a circle of radius r, an arc subtending an angle θ (in radians) at the centre has length l = rθ. If θ is in degrees, l = (θ/360) × 2πr.

Formula: l = rθ

Rule 1: Arc length from degrees — Convert the angle to radians first.

  • Easy: r = 6 cm, θ = 60°: θ = π/3, l = 2π cm.
  • Medium: r = 21 cm, θ = 30°: l = 21 × π/6 = 11 cm (π = 22/7).
  • Exam-type: r = 9 m, θ = 90°: l = 4.5π m.

Rule 2: Find the angle or radius — Rearrange l = rθ.

  • Easy: l = 16 cm, r = 8 cm: θ = 2 radians = 114.6°.
  • Medium: l = 11 cm, θ = π/2, π = 22/7: r = 7 cm.
  • Exam-type: l = 5π cm, r = 15 cm: θ = π/3 = 60°.
Remember

A longer radius gives a longer arc for the same angle.

Common mistake

Using degrees directly in l = rθ.

Memory clue

Radian angle × radius = arc.

Practice questions

Easy
  1. 1r = 10 cm, θ = 0.6 radian. Find l.
  2. 2r = 12 cm, θ = π/6. Find l.
  3. 3l = 20 cm, r = 5 cm. Find θ in radians.
Medium
  1. 4r = 14 cm, θ = 45°. Find l (π = 22/7).
  2. 5A pendulum of length 1 m swings through 0.2 radian. How far does the bob travel along its path?
  3. 6A wheel of radius 35 cm turns through 2 radians. How far does a point on its rim move?
Challenging
  1. 7An arc of length 22 cm on a circle of radius 14 cm subtends how many degrees? (π = 22/7)
  2. 8The hour hand is 5 cm long. How far does its tip travel in 4 hours? (π = 3.14)
Show answers (try them yourself first)
  1. 16 cm
  2. 22π cm
  3. 34
  4. 4l = 14 × π/4 = 11 cm
  5. 50.2 m
  6. 670 cm
  7. 7θ = 22/14 radian = 1.571 = π/2, so 90°
  8. 8Angle = 4 × 30° = 120° = 2π/3; l = 5 × 2.0933 ≈ 10.47 cm