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State Board · Class 11 · Mathematics Part 1

Lesson 1: Angle and its Measurement

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी

Directed angle

From daily life

To say how far a clock hand has moved we give both "how much" and "which way". "Turn right" and "turn left" are also direction plus angle. Building on classes 9–10 : There we met angles from 0° to 180° and the angles of a triangle; now angles carry a direction and can also be larger than 360°.

Simple definition

Keep one ray fixed and rotate another ray about the common point: the angle formed is a directed angle (दिशायुक्त कोन). The starting ray is the initial ray and the ray where the rotation stops is the terminal ray. Turning against the clock gives a positive angle; turning with the clock gives a negative angle.

Types of angles : If the ray does not turn at all it is the zero angle (0°); a quarter turn is a right angle (90°); a half turn is a straight angle (180°); a full turn is 360°. The ordered pairs of rays (OA, OB) and (OB, OA) are different, so ∠AOB and ∠BOA are not the same directed angle.

Rule 1: Standard position and quadrants — An angle is in standard position when its initial ray lies along the positive X-axis and its vertex is the origin O. The quadrant of the angle is the one that contains the terminal ray: 0° to 90° is quadrant I, 90° to 180° is II, 180° to 270° is III, 270° to 360° is IV. For example 150° lies in quadrant II, 250° in III and −60° in IV.

OABθIIIIIIIVXY

In the figure : θ = ∠XOB is in standard position; OA is the initial ray and OB the terminal ray. Here θ = 135°, so it lies in quadrant II.

  • Easy : 30° is in quadrant I and 120° is in quadrant II.
  • Medium : 300° is in quadrant IV; −120° is 120° clockwise, which lies in quadrant III.
  • Exam-type : −200° is 200° clockwise, so the terminal ray is at 160°: quadrant II.

Rule 2: Quadrantal angles — If the terminal ray lies on the X-axis or the Y-axis the angle is a quadrantal angle: 0°, 90°, 180°, 270°, 360° and their negative or larger values (−90°, 450° and so on). It lies in no quadrant.

  • Easy : 90° and 180° are quadrantal angles.
  • Medium : −90° is quadrantal (the terminal ray is on the negative Y-axis).
  • Exam-type : 540° = 360° + 180°, the terminal ray is on the negative X-axis, so it is quadrantal.

Rule 3: Co-terminal angles — Directed angles with the same initial ray and the same terminal ray are co-terminal angles. The co-terminal angles of θ are θ + 360° × n (n any integer). So the difference of two co-terminal angles is a whole multiple of 360°.

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Directed angle

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  • Easy : 60° and 420° are co-terminal because 420° − 60° = 360°.
  • Medium : 45° and −315° are co-terminal because 45° − (−315°) = 360°.
  • Exam-type : Co-terminal angles of 60°: 420°, 780°, −300°, −660°.

How is the formula obtained? : After a full turn (360°) the ray comes back to the same place. So adding or subtracting 360° any number of times does not move the terminal ray: co-terminal angles = θ + 360° × n. Two such angles θ + 360°n₁ and θ + 360°n₂ differ by 360° × (n₁ − n₂), a whole multiple of 360°.

Remember

Positive angle = against the clock; negative angle = with the clock. Co-terminal angles differ by a multiple of 360°.

Common mistake

Calling 90° "quadrant I or II". It is a quadrantal angle and lies in no quadrant.

Shortcut

To find the quadrant of a large angle, divide by 360° and look at the remainder: 1000° → 1000 − 720 = 280° → quadrant IV. If the remainder of a negative angle is negative, add 360° to it. Memory clue : "With the clock hands = minus, against them = plus"; quadrants I → II → III → IV run against the clock in steps of 90°.

CET trap

An option may call 90° or 180° "quadrant I" or "quadrant II", but those are quadrantal angles. Also do not stop at a negative remainder: −1110° reduces to −30°, so find the quadrant from 330°.

CET-style example : In which quadrant does −1110° lie? (A) I (B) II (C) III (D) IV

Answer : (D). 1110 = 3 × 360 + 30, so −1110° behaves like −30°; −30° + 360° = 330° → quadrant IV. Time-saving method : The nearest multiple of 360 is 1080; 1110 − 1080 = 30 → −30°, a small turn with the clock hands, hence quadrant IV.

Spot the difference

AngleTerminal rayQuadrant / type
45°in quadrant IQuadrant I
405°same place as 45° (one extra full turn)Quadrant I; co-terminal with 45°
−45°45° with the clockQuadrant IV
90°on the positive Y-axisQuadrantal; in no quadrant
−270°on the positive Y-axisQuadrantal; co-terminal with 90°

Practice questions

Easy
  1. 1In a directed angle, which is the initial ray?
  2. 2In which quadrant is 210°?
  3. 3Write one co-terminal angle of 45°.
Medium
  1. 4In which quadrant is −30°?
  2. 5Find the co-terminal angle of 400° between 0° and 360°.
  3. 6What kind of angle is 270°?
Challenging
  1. 7Find the co-terminal angle of −780° between 0° and 360°, and its quadrant.
  2. 8Are 120° and −240° co-terminal?
Show answers (try them yourself first)
  1. 1The ray from which the rotation starts.
  2. 2Quadrant III (180° to 270°).
  3. 3405° (or −315°).
  4. 4Quadrant IV (−30° ≡ 330°).
  5. 540° (400° − 360°).
  6. 6A quadrantal angle (on the negative Y-axis).
  7. 7300°, quadrant IV (−780° + 1080° = 300°).
  8. 8Yes, 120° − (−240°) = 360°.

Degree measure: degrees, minutes, seconds

From daily life

An hour has 60 minutes and a minute has 60 seconds. The degree system for angles also works in groups of 60. Building on classes 9–10 : There we measured angles in whole degrees; now we use the smaller parts of a degree (minutes and seconds) for exact work.

Simple definition

One degree (1°) is the 360th part of a full turn. The 60th part of 1° is 1 minute (1′) and the 60th part of 1′ is 1 second (1″). This is the sexagesimal system (षष्ठिक पद्धत).

Rule 1: Conversion — 1° = 60′, 1′ = 60″, so 1° = 3600″. A right angle = 90°, a straight angle = 180°, a full turn = 360°. Multiply the decimal part of a degree by 60 to get minutes, and the decimal part of the minutes by 60 to get seconds.

  • Easy : 25°30′ = 25.5° because 30′ = 30/60 = 0.5°.
  • Medium : 52.35° = 52° + 0.35 × 60′ = 52°21′.
  • Exam-type : 37.46° = 37° + 0.46 × 60′ = 37°27.6′ = 37°27′36″ (0.6 × 60 = 36).

How is the formula obtained? : Since 1° = 60′, 0.35° = 0.35 × 60′ = 21′ (just as 0.35 hour = 0.35 × 60 = 21 minutes). Multiply the leftover decimal of the minutes by 60 again to get seconds, because 1′ = 60″. In the other direction: seconds ÷ 60 = minutes, minutes ÷ 60 = degrees.

1°60′3600″×60÷60×60÷60

In the figure : Going from degrees to minutes to seconds, multiply by 60 at each step; coming back, divide by 60.

Rule 2: Adding and subtracting D-M-S — When seconds reach 60 make 1′; when minutes reach 60 make 1°. In subtraction borrow 1° as 60′.

  • Easy : 20°40′ + 10°30′ = 30°70′ = 31°10′.
  • Medium : 90° − 38°25′ = 89°60′ − 38°25′ = 51°35′.
  • Exam-type : 36°45′30″ = 36 + 45/60 + 30/3600 = 36.7583…°.
Remember

Minutes and seconds here measure angles; they are not the minutes and seconds of time.

Common mistake

Writing 25.5° as 25°50′. Since 0.5° = 30′, it is 25°30′.

Shortcut

Decimal degree → minutes: decimal × 60. Minutes → degrees: minutes ÷ 60. Memory clue : Think of a clock! Degree = hour, ′ = minute, ″ = second: the same groups of 60, only the signs differ.

CET trap

An option such as 52°35′ for 52.35° is always present. The decimal part is not the minutes itself; multiply by 60: 0.35 × 60 = 21′, so 52°21′.

CET-style example 1 : 12.7625° = ? (A) 12°45′45″ (B) 12°46′15″ (C) 12°45′25″ (D) 12°47′45″

Answer : (A). 0.7625 × 60 = 45.75′ = 45′ + 0.75 × 60″ = 45′45″. Time-saving method : Recall 0.75° = 45′ at once; the remaining 0.0125° × 3600 = 45″.

CET-style example 2 : 180° − 64°38′52″ = ? (A) 115°21′8″ (B) 115°22′8″ (C) 116°21′8″ (D) 115°21′52″

Answer : (A). 180° = 179°59′60″: 179 − 64 = 115, 59 − 38 = 21, 60 − 52 = 8. Time-saving method : Write 180° at once as 179°59′60″; then no borrowing is needed.

Spot the difference

MeasureIn degreesD-M form
25°30′25.5°25°30′
52.35°52.35°52°21′
37.46°37.46°37°27′36″

Minute of angle and minute of time

PointAngle (degree system)Time (clock)
Big unit1 degree (°)1 hour
Middle unit1′ = 60th part of a degree1 minute = 60th part of an hour
Small unit1″ = 60th part of a minute1 second = 60th part of a minute
Signs° ′ ″h min s
Measuresa turn (with direction)duration

Practice questions

Easy
  1. 1How many minutes are in 1°?
  2. 2Write 40°30′ in degrees.
  3. 3Write 15.5° in degrees and minutes.
Medium
  1. 4Write 63.45° in degrees and minutes.
  2. 535°40′ + 47°35′ = ?
  3. 6How many seconds are in 1°?
Challenging
  1. 7120° − 47°18′ = ?
  2. 8Write 36°45′ as a decimal of a degree.
Show answers (try them yourself first)
  1. 160′.
  2. 240.5°.
  3. 315°30′.
  4. 463°27′ (0.45 × 60 = 27).
  5. 583°15′ (35°40′ + 47°35′ = 82°75′ = 83°15′).
  6. 63600″.
  7. 772°42′ (119°60′ − 47°18′).
  8. 836.75°.

Radian measure

From daily life

Bend a piece of string along a circle: when the string is as long as the radius, the angle it makes is 1 radian. It links length and angle. Building on classes 9–10 : There we measured angles in degrees and used (θ/360) × 2πr for an arc; now we measure the angle "in radius units", which makes the formulas much shorter.

Simple definition

The angle at the centre of a circle subtended by an arc equal in length to the radius measures 1 radian (1c). The radian does not depend on the size of the circle, so it is well defined. θ = s ÷ r.

Orrs = r1c

In the figure : The green arc has s = r; the angle it subtends is 1c (≈ 57.3°).

Rule 1: πc = 180° — The circumference is 2πr, so a full turn = 2π radians = 360°. Hence πc = 180° and 1c = 180°/π ≈ 57.3°.

  • Easy : 180° = π radians, 360° = 2π radians.
  • Medium : 90° = π/2, 60° = π/3, 30° = π/6, 45° = π/4.
  • Exam-type : 270° = 3π/2 and 120° = 2π/3.

How is πc = 180° obtained? :

  • 1) A central angle is proportional to the length of its arc. For 1 radian the arc is r.
  • 2) A semicircle has arc πr, and its angle is a straight angle, 180°.
  • 3) So (straight angle) ÷ (1 radian) = πr ÷ r = π.
  • 4) Hence a straight angle = π radians, that is πc = 180° and 1c = 180°/π.

Rule 2: Conversion formula — Degrees → radians: degrees × π/180. Radians → degrees: radians × 180/π. The sign of a negative angle stays the same.

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Radian measure

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  • Easy : 60° × π/180 = π/3 radians.
  • Medium : −150° = −150 × π/180 = −5π/6; 7π/4 radians = 7 × 180/4 = 315°.
  • Exam-type : 50°15′ = 50.25° = 50.25 × π/180 = 67π/240 radians.

How is the conversion formula obtained? : If one angle is θ degrees and x radians, it is the same fraction of a straight angle in both: θ/180 = x/π. Hence x = θ × π/180 and θ = x × 180/π.

Useful values : 1° = π/180 ≈ 0.01745 radian. 1c = 180°/π ≈ 57.2958° ≈ 57°17′45″ (using the exact value of π). With π = 3.14 you get 57.3248° = 57°19′29″, so use the π given in the question.

Remember

If no unit is written the angle is taken in radians: sin 2 means 2 radians.

Common mistake

Treating π as 180. The number π is about 3.14; it is the angle πc (π radians) that equals 180°.

Shortcut

To remember the radians of 30°, 45°, 60°, 90°: divide 180° by 6, 4, 3, 2 → π/6, π/4, π/3, π/2. Memory clue : "Put the wanted one on top!" — for degrees → radians the radian sign π goes on top: × π/180. For radians → degrees the 180 goes on top: × 180/π.

CET trap

A radian answer appears in the options in lowest terms (7π/12), never as 105π/180. A number written without π (such as 2 or 1.5) is in radians; and in arc or area questions convert θ to radians first if it is given in degrees.

CET-style example : 7π/15 radians = ? degrees (A) 84° (B) 96° (C) 72° (D) 105°

Answer : (A). π/15 = 180°/15 = 12°, so 7π/15 = 7 × 12° = 84°. Time-saving method : Write 180° in place of π and divide by the denominator: 180 ÷ 15 = 12, then × 7 = 84.

Spot the difference

DegreesRadians
15°π/12
30°π/6
45°π/4
60°π/3
90°π/2
120°2π/3
135°3π/4
150°5π/6
180°π
270°3π/2
360°2π

Degrees and radians compared

PointDegree (°)Radian (c)
Based on360th part of a full turnan arc equal to the radius
Right angle90°π/2
Straight angle180°π
Full turn360°2π
Size of 1 unit1° ≈ 0.01745 radian1 radian ≈ 57.3°
Writing the unit° is neededc, or nothing

Practice questions

Easy
  1. 1180° in radians?
  2. 2π/2 radians in degrees?
  3. 345° in radians?
Medium
  1. 4−135° in radians?
  2. 57π/6 radians in degrees?
  3. 6About how many degrees is 1 radian?
Challenging
  1. 72 radians in degrees (take π = 22/7).
  2. 8100° in radians?
Show answers (try them yourself first)
  1. 1π.
  2. 290°.
  3. 3π/4.
  4. 4−3π/4.
  5. 5210°.
  6. 6About 57.3°.
  7. 72 × 180 × 7/22 ≈ 114.5° (exact value 114.59°).
  8. 85π/9.

Angles of triangles, polygons and the clock

From daily life

The gap between the two hands of a clock keeps changing. A triangular plate or a hexagonal tile also has angles that fit a fixed rule. Building on classes 9–10 : There we learnt that the angles of a triangle add up to 180° and met the angles of a quadrilateral; now we link the angles of any polygon with radians and with the clock.

Simple definition

The three angles of a triangle add up to 180° (π radians). The interior angles of a polygon with n sides add up to (n − 2) × 180°. Each angle of a regular polygon is (n − 2) × 180°/n.

Rule 1: Triangles and polygons — A quadrilateral has angle sum 360°, a pentagon 540°, a hexagon 720°. Each angle of a regular hexagon is 720°/6 = 120°; of a regular octagon 1080°/8 = 135°. The exterior angle of a regular polygon is 360°/n.

  • Easy : A triangle has two angles 40° and 60°; the third is 180° − 100° = 80°.
  • Medium : A triangle with angles 40°, 60°, 80° has 2π/9, π/3, 4π/9 radians; the sum is π.
  • Exam-type : Triangle angles in the ratio 1 : 2 : 3; k = 180°/6 = 30°, so the angles are 30°, 60°, 90°.

How is the formula obtained? : From one vertex of an n-sided polygon draw lines to the other vertices: this makes (n − 2) triangles. Each has angle sum 180°, so the total is (n − 2) × 180°. For a regular polygon each interior angle = (n − 2) × 180°/n and the exterior angle = 180° − interior angle = 360°/n (all exterior angles add up to 360°).

Rule 2: Angles on a clock — The minute hand turns 360° in 60 minutes: 6° per minute. The hour hand turns 360° in 12 hours: 30° per hour and 0.5° per minute. So at h hours m minutes the angle between the hands is |30h − 5.5m|.

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Angles of triangles, polygons and the clock

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  • Easy : At 3:00 the angle is 30 × 3 = 90°.
  • Medium : In 15 minutes the minute hand turns 15 × 6 = 90° and the hour hand 7.5°.
  • Exam-type : At 3:30 the angle is |30 × 3 − 5.5 × 30| = |90 − 165| = 75°.

How is the formula obtained? : After h hours m minutes the hour hand is 30h + 0.5m degrees from 12 and the minute hand is 6m degrees. The difference is (30h + 0.5m) − 6m = 30h − 5.5m, and an angle between hands is taken as positive, so use its positive value. If the answer is more than 180°, the smaller angle = 360° − answer.

1236970°H = 5×30 + 40×0.5 = 170°M = 40×6 = 240°240° − 170° = 70°

In the figure : At 5:40 the hour hand is 5 × 30 + 40 × 0.5 = 170° from 12 and the minute hand is 40 × 6 = 240° from 12; the difference is 240° − 170° = 70°.

Rule 3: Number of sides of a regular polygon — If the interior angle is given, first find the exterior angle = 180° − interior angle, then n = 360° ÷ exterior angle. If the interior angle is in radians, convert it to degrees first.

  • Easy : Interior angle 135°: exterior angle 45°, n = 360 ÷ 45 = 8.
  • Medium : Interior angle 3π/5 radians = 108°: exterior angle 72°, n = 360 ÷ 72 = 5.
  • Exam-type : Interior angle 9π/10 radians = 162°: exterior angle 18°, n = 360 ÷ 18 = 20.

Rule 4: Angles in a ratio or in an arithmetic progression — If the angles are in a ratio (for example 2 : 3 : 4), take them as 2k, 3k, 4k and use the angle sum (triangle 180°, quadrilateral 360°). If the angles of a triangle are in an arithmetic progression (A.P.), take them as (a − d), a, (a + d); the sum is 3a = 180°, so the middle angle is a = 60°.

  • Easy : The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4; k = 360° ÷ 10 = 36°, so the angles are 36°, 72°, 108°, 144°.
  • Medium : The angles of a triangle are in A.P. and the smallest is 20°: a = 60°, a − d = 20° so d = 40°; the angles are 20°, 60°, 100°.
  • Exam-type : The angles of a triangle are in A.P. and the largest is 3 times the smallest: a − d = k, a + d = 3k so a = 2k = 60°, k = 30°; the angles are 30°, 60°, 90° (π/6, π/3, π/2 radians).
Remember

Minute hand 6° per minute; hour hand 0.5° per minute.

Common mistake

Keeping the hour hand exactly on 3 at 3:30. It is halfway between 3 and 4, at 105°.

Shortcut

Angle sum of a polygon = (n − 2) × 180°. Split it into n − 2 triangles. Memory clue : On a clock, "6 minus half = five and a half": the minute hand moves 6° and the hour hand 0.5° per minute, so 5.5m. And "n − 2 = number of triangles" (draw lines from one vertex).

CET trap

If the "smaller angle" is asked and 30h − 5.5m gives more than 180°, that very value is in the options, but the correct answer is 360° − value. Also check whether the question gives the interior or the exterior angle; both are in the options.

CET-style example 1 : What is the smaller angle between the hands of a clock at 5:40? (A) 70° (B) 80° (C) 100° (D) 110°

Answer : (A). 30 × 5 − 5.5 × 40 = 150 − 220 = −70, so 70°. Time-saving method : The minute hand at 40 is at 40 × 6 = 240°; the hour hand, 40 minutes past 5, is at 150° + 20° = 170°; the difference is 70°.

CET-style example 2 : In a regular polygon the ratio of an interior angle to an exterior angle is 5 : 1. How many sides does it have? (A) 10 (B) 12 (C) 8 (D) 15

Answer : (B). Interior + exterior = 180°, so exterior = 180° ÷ 6 = 30° and n = 360° ÷ 30° = 12. Time-saving method : Divide 180° by the sum of the ratio parts (5 + 1 = 6); then 360 ÷ exterior angle.

Spot the difference

FigurenAngle sumInteriorExterior
Regular triangle3180°60°120°
Square4360°90°90°
Regular pentagon5540°108°72°
Regular hexagon6720°120°60°
Regular octagon81080°135°45°
Regular decagon101440°144°36°

Clock hands compared

HandIn 1 minuteIn 1 hourIn 12 hours
Minute hand6°360°4320°
Hour hand0.5°30°360°

Practice questions

Easy
  1. 1What is the sum of the angles of a triangle?
  2. 2Each angle of a regular pentagon?
  3. 3The angle between the clock hands at 4:00?
Medium
  1. 4Angle sum of an octagon?
  2. 5Triangle angles in the ratio 2 : 3 : 4; find them in degrees.
  3. 6How many degrees does the minute hand turn in 20 minutes?
Challenging
  1. 7The angle between the hands at 9:30.
  2. 8A regular polygon has each angle 150°. How many sides?
Show answers (try them yourself first)
  1. 1180°.
  2. 2108° (540°/5).
  3. 3120°.
  4. 41080°.
  5. 540°, 60°, 80° (k = 180°/9 = 20°).
  6. 6120°.
  7. 7105° (hour hand 285°, minute hand 180° → 105°).
  8. 812 (exterior angle 30° = 360°/n).

Arc length and area of a sector

From daily life

A slice of pizza is a sector and its curved edge is the arc. The wider the slice (larger angle), the longer the edge and the bigger the slice. Building on class 10 : There, in mensuration, we found the arc and the sector with the (θ/360) formulas in degrees; with radians the same formulas shrink to s = rθ and A = ½r²θ.

Simple definition

In a circle of radius r with central angle θ (in radians), the arc length is s = rθ and the area of the sector is A = ½r²θ = ½rs.

OrθsA

In the figure : The orange part is the sector A, the green arc is s and the central angle is θ.

How is the formula obtained? : Both the arc and the area are proportional to the central angle. A full circle has angle 2π radians and circumference 2πr: s/(2πr) = θ/(2π), so s = rθ. The whole circle has area πr²: A/(πr²) = θ/(2π), so A = ½r²θ. Putting s = rθ gives A = ½r × (rθ) = ½rs.

Rule 1: Arc length — s = rθ with θ in radians. If θ is given in degrees, first multiply by π/180.

  • Easy : r = 7 cm, θ = π/2: s = 7 × π/2 = 11 cm (π = 22/7).
  • Medium : r = 6 cm, θ = 60° = π/3: s = 6 × π/3 = 2π ≈ 6.28 cm.
  • Exam-type : The tip of a 10 cm minute hand travels 10 × 2π/3 ≈ 20.94 cm in 20 minutes (θ = 120°).

Rule 2: Area of a sector — A = ½r²θ = ½rs. Doubling θ doubles the area. In degrees: A = (θ°/360°) × πr².

Live figure

Arc length and area of a sector

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  • Easy : r = 7, θ = π/2: A = ½ × 49 × π/2 = 38.5 cm² (π = 22/7).
  • Medium : r = 6, θ = π/3: A = ½ × 36 × π/3 = 6π ≈ 18.85 cm².
  • Exam-type : r = 10, s = 15: A = ½ × 10 × 15 = 75 cm² and θ = s/r = 1.5 radians.

Rule 3: Perimeter of a sector — The boundary of a sector is two radii and one arc. Perimeter P = 2r + s = r(2 + θ) (θ in radians). The arc alone is not the perimeter.

  • Easy : r = 10 cm, θ = 1.2 radians: s = 12 cm, P = 20 + 12 = 32 cm.
  • Medium : r = 9 cm, θ = 40° = 2π/9: s = 2π ≈ 6.28 cm, P = 18 + 2π ≈ 24.28 cm.
  • Exam-type : r = 9 cm, P = 36 cm: 18 + s = 36 so s = 18 cm and θ = 18 ÷ 9 = 2 radians.

Rule 4: Area of a segment of a circle — The part of a circle between a chord and its arc is a segment of a circle. Area of the segment = area of the sector − area of triangle OPQ = ½r²θ − ½r² sin θ = ½r²(θ − sin θ). The values of sin (30°, 60°, 90°) are from class 10: ½, √3/2, 1.

  • Easy : r = 10 cm, θ = 90° = π/2: sector = ½ × 100 × π/2 = 25π; triangle = ½ × 100 × 1 = 50; segment = 25π − 50 ≈ 28.54 cm².
  • Medium : r = 8 cm, θ = 30° = π/6: sector = ½ × 64 × π/6 = 16π/3 ≈ 16.76; triangle = ½ × 64 × ½ = 16; segment ≈ 0.76 cm².
  • Exam-type : r = 10 cm, θ = 60° = π/3: sector = 50π/3 ≈ 52.36; triangle = ½ × 100 × √3/2 = 25√3 ≈ 43.30; segment ≈ 9.06 cm².
θOPQrrA₁ = ½r²θA₂ = ½r² sin θA₁ − A₂

In the figure : The green part is the segment; its area = (sector A₁ with the orange dashed boundary) − (blue triangle A₂). Here θ = 60°.

Remember

Use s = rθ and A = ½r²θ only when θ is in radians.

Common mistake

Using s = rθ with θ in degrees. Convert to radians first.

Shortcut

If θ and r are known, find s and then A = ½ × r × s: two steps. Memory clue : A sector looks like a triangle: base = arc s, height = radius r; so A = ½ × base × height = ½rs.

CET trap

When the "perimeter" is asked, an option with just the arc s is always present; when the "area" is asked, an option with r²θ (the ½ forgotten) is always present. Also, when the radius changes the arc changes in proportion to r, but the area changes in proportion to r².

CET-style example 1 : What is the length of the arc subtended by a 36° central angle in a circle of radius 35 cm? (take π = 22/7) (A) 11 cm (B) 22 cm (C) 44 cm (D) 7 cm

Answer : (B). 36° is 1/10 of a full turn, so s = (1/10) × 2 × 22/7 × 35 = 22 cm. Time-saving method : Since 36° = 360°/10, skip radians and take one tenth of the circumference directly.

CET-style example 2 : The central angle of a sector is kept the same and the radius is increased by 20%. By what percent does the area increase? (A) 20% (B) 40% (C) 44% (D) 48%

Answer : (C). A ∝ r²: (6/5)² = 36/25 = 1.44, an increase of 44%. Time-saving method : A 20% increase = × 6/5; square it: 36/25.

Spot the difference

QuantityIn degrees (θ°)In radians (θ)Unit
Arc length s(θ°/360°) × 2πrrθcm
Sector area A(θ°/360°) × πr²½r²θ = ½rscm²
Sector perimeter P2r + s2r + rθcm
Segment areasector − triangle½r²(θ − sin θ)cm²
Central angle θ—s/rradians

Practice questions

Easy
  1. 1r = 5 cm, θ = 2 radians: find s.
  2. 2s = 12 cm, r = 4 cm: find θ.
  3. 3r = 10 cm, s = 6 cm: find A.
Medium
  1. 4r = 12 cm, θ = π/6: find s.
  2. 5r = 6 cm, θ = 60°: find A.
  3. 6s = 15 cm, r = 10 cm: θ in degrees (π = 3.14).
Challenging
  1. 7s = 22 cm, θ = π/2: find r (π = 22/7).
  2. 8r = 8 cm, θ = 1.5 radians: find s and A.
Show answers (try them yourself first)
  1. 110 cm.
  2. 23 radians.
  3. 3½ × 10 × 6 = 30 cm².
  4. 42π ≈ 6.28 cm.
  5. 56π ≈ 18.85 cm².
  6. 61.5 radians ≈ 85.99° (1.5 × 180/3.14).
  7. 7r = 22 ÷ (π/2) = 44/π = 14 cm.
  8. 8s = 12 cm, A = ½ × 64 × 1.5 = 48 cm².

Quick revision of the whole lesson

ConceptKey formula / factUnit / note
Co-terminal anglesθ + 360° × ndifference is a multiple of 360°
Quadrants0°–90° I, 90°–180° II, 180°–270° III, 270°–360° IV90°, 180°… are quadrantal
Degree-minute-second1° = 60′ = 3600″decimal × 60
Degree ↔ radianπc = 180°; × π/180; × 180/π1c ≈ 57.3°
Polygonangle sum (n − 2) × 180°; exterior angle 360°/nn = 360° ÷ (180° − interior angle)
Clock30h − 5.5m (positive value)if more than 180°, 360° − answer
Arc lengths = rθθ in radians
Sector areaA = ½r²θ = ½rscm²
Sector perimeterP = 2r + scm
Segment area½r²(θ − sin θ)cm²

🎯 Most important points for the exam

  • Positive angle = against the clock; negative angle = with the clock; a quadrantal angle lies in no quadrant.
  • Quadrant of a large angle: add or subtract multiples of 360° to bring it between 0° and 360°.
  • Decimal degree × 60 = minutes; leftover decimal × 60 = seconds.
  • Degrees → radians: × π/180; radians → degrees: × 180/π ("put the wanted one on top").
  • Sides of a regular polygon: exterior angle = 180° − interior angle, n = 360° ÷ exterior angle.
  • Clock angle: 30h − 5.5m; if the answer is more than 180°, subtract it from 360°.
  • s = rθ and A = ½r²θ only in radians; if degrees are given, convert first.
  • Sector perimeter = 2r + s; segment = sector − triangle.
Study mantra

directed angle → co-terminal → degree-minute-second → radian → polygon and clock → s = rθ → A = ½r²θ → perimeter and segment.