Sector, Segment and Combined Figures
Sector, Segment and Combined Figures
A sector is a slice of a circle cut by two radii; a segment is the part cut off by a chord. Both are fractions of a full circle by the angle θ/360. NTPC CBT-2 extends this to shaded regions built from squares, triangles and circular arcs.
1. Formula Box
Arc length = (θ/360) × 2πr Sector area = (θ/360) × πr² = (1/2) × arc × r
Sector perimeter = arc + 2r
Segment area = sector area − triangle area = (θ/360)πr² − (1/2) r² sin θ
| θ | Fraction of circle | Triangle area in the segment formula |
|---|---|---|
| 60° | 1/6 | (√3/4) r² (equilateral) |
| 90° | 1/4 | r²/2 |
| 120° | 1/3 | (√3/4) r² |
| 180° | 1/2 | 0 (semicircle) |
Clock link: the minute hand sweeps 6° per minute; the hour hand 0.5° per minute.
✗ Sector perimeter = arc length | ✓ Add both radii: arc + 2r
✗ Segment area = sector area | ✓ Subtract the triangle formed by the two radii and the chord
हिंदी नोट: त्रिज्यखंड का क्षेत्रफल = (θ/360) × πr² और चाप की लंबाई = (θ/360) × 2πr होती है। वृत्तखंड (segment) = त्रिज्यखंड − त्रिभुज।
Exam Pointer: Sector, segment and shaded-region questions had no verified NTPC paper label in our check, so they are tagged syllabus-based. They are standard Class 10 mensuration and appear more in CBT-2-level practice; the CBT-2 block below covers shaded regions.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Sector area and arc length
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find the arc length and area of a sector of radius 21 cm and angle 60°.
Arc = (1/6) × 132 = 22 cm. Area = (1/6) × 1,386 = 231 cm².
Answer: 22 cm; 231 cm²
EXAMATLAS LEVEL
Q. The perimeter of a sector of radius 14 cm is 50 cm. Find its angle and area.
Arc = 50 − 28 = 22 cm. Full circumference = 88 cm, so θ = 22/88 × 360 = 90°. Area = (1/4) × 616 = 154 cm² (also (1/2) × 22 × 14).
Answer: 90°; 154 cm²
Pattern 2: Segment of a circle
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find the area of the minor segment of a circle of radius 14 cm cut off by a chord subtending 90° at the centre.
Sector = 154 cm²; triangle = (1/2) × 14 × 14 = 98 cm². Segment = 56 cm².
Answer: 56 cm²
EXAMATLAS LEVEL
Q. A chord of a circle of radius 12 cm subtends 60° at the centre. Find the area of the minor segment (use π = 3.14, √3 = 1.73).
Sector = (1/6) × 3.14 × 144 = 75.36 cm². The triangle is equilateral with side 12: (1.73/4) × 144 = 62.28 cm². Segment = 13.08 cm² (exactly 24π − 36√3 ≈ 13.04).
Answer: About 13.08 cm²
CBT-2 LEVEL
Pattern 3: Shaded regions from squares, triangles and arcs
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Four quadrants of radius 7 cm are drawn at the corners of a square of side 14 cm. Find the area of the square not covered by them.
The four quadrants make one full circle of radius 7: 154 cm². Shaded = 196 − 154 = 42 cm².
Answer: 42 cm²
EXAMATLAS LEVEL
Q. Circles of radius 6 cm are drawn with centres at the vertices of an equilateral triangle of side 12 cm. Find the area of the triangle not covered by the circles (π = 3.14, √3 = 1.73).
Each vertex angle is 60°, so the three sectors inside the triangle total 180°, half a circle: (1/2) × 3.14 × 36 = 56.52 cm². Triangle = (1.73/4) × 144 = 62.28 cm². Uncovered = 5.76 cm² (exactly 36√3 − 18π ≈ 5.81).
Answer: About 5.76 cm²
Pattern 4: Clock hands: arc travelled and area swept
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. The minute hand of a clock is 21 cm long. How far does its tip move in 20 minutes?
In 20 minutes the minute hand turns 120°, one-third of a circle. Distance = (1/3) × 2 × 22/7 × 21 = 44 cm.
Answer: 44 cm
EXAMATLAS LEVEL
Q. Find the area swept by a 14 cm minute hand between 8:10 and 8:40, and by a 7 cm hour hand in the same time.
30 minutes: the minute hand turns 180°, sweeping (1/2) × 616 = 308 cm². The hour hand turns 0.5° per minute, so 15°, sweeping (15/360) × 154 = 6 5/12 cm² (about 6.42 cm²).
Answer: 308 cm²; about 6.42 cm²
60-Second Revision
- Arc = (θ/360)2πr; sector = (θ/360)πr² = (1/2) arc × r; perimeter = arc + 2r.
- Segment = sector − triangle; 90° triangle = r²/2, 60° triangle = (√3/4)r².
- Equal-radius sectors at the vertices of any triangle add up to 180°, i.e. half a circle.
Next Step: Mensuration 2D done. Practise these in the ExamAtlas RRB NTPC 2026 mock tests; the same formulas feed directly into the 3D solids chapter.