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Areas related to circles

Lesson 9 of 113

1. Ye topic hai kya?

Circle ka hissa — poora nahi to kitna? Paper: r=14 pe area, 60° sector, chord ke baahar ka segment, do concentric (ring), square mein inscribed circle ka leftover. θ degree (Class-X). π = 22/7 jab r 7 ka multiple; warna 3.14.

Yahan lock

C=2πr, A=πr², sector (θ/360)πr², arc (θ/360)2πr, segment = sector − triangle, ring π(R²−r²).

Yahan kya nahi

4πr² sphere (3D agla). Tangent ⊥, PA=PB. Diameter-OP construction. Tower 30°. Coordinate x²+y²=r².

2. Poora circle — C aur A

Circumference C = 2πr = πd. Area A = πr². Diameter d=2r — area mein r², d mat square without /4. (πd²/4 same as πr².)
Example

r = 14 cm, π = 22/7. C aur A?

Solution

  1. C

    2 × 22/7 × 14 = 88 cm.

  2. A

    22/7 × 196 = 616 cm². Trap: 4πr² = 2464 (sphere). 2πr² mix.

Answer88 cm, 616 cm²

r=35, 4 rounds around: 4 × 2 × 22/7 × 35 = 880. (Park wire — 3D nahi.)

3. Sector aur arc

Sector = pizza slice: do radii + beech ka arc. Central angle θ. Fraction of whole = θ/360.

Area of sector = (θ/360) × πr². Length of arc = (θ/360) × 2πr. Quadrant θ=90° → 1/4. Semicircle 180° → 1/2. θ>180 major sector; paper usually minor unless “major” likha.
O 90° sector

Highlight = 90° sector. Poora circle cream. Arc = curved edge only, radii nahi.

Example

r = 21, θ = 60°, π = 22/7. Sector area aur arc length?

Solution

  1. Fraction

    60/360 = 1/6.

  2. Area

    (1/6) × 22/7 × 441 = (1/6) × 22 × 63 = 231 cm².

  3. Arc

    (1/6) × 2 × 22/7 × 21 = (1/6) × 132 = 22 cm. Same 22 as r=14 ka 90° arc — coincidence, formula alag.

Answer231 cm², 22 cm
Arc length ≠ sector area. θ radians XI. Perimeter of sector = arc + 2r (do radii). Sirf arc mat bolna “sector ka perimeter”.

4. Segment — sector minus triangle

Chord AB circle ko do pieces: chhota minor segment, bada major. Minor = sector O-AXB − ΔOAB.

Minor segment area = sector − Δ (do radii + chord). 90°: triangle isosceles right, ½ r². 60°: equilateral, (√3/4) r². Major segment = πr² − minor.
O segment

Shaded cap = minor segment (90°). Triangle O–chord alag; usko sector se ghatana.

Example

r = 14, θ = 90°, π = 22/7. Minor segment?

Solution

  1. Sector

    (90/360)×616 = 154.

  2. Δ

    ½ × 14 × 14 = 98. (Right isosceles, legs radii.)

  3. Minus

    154 − 98 = 56 cm². Sirf 154 sector hai, segment nahi.

Answer56 cm²
Example

r = 14, θ = 60°. Minor segment formula?

Solution

  1. Sector

    (60/360)×πr² = πr²/6. π=22/7 → 616/6 = 308/3.

  2. Δ

    60° + do radii r ⇒ equilateral. Area (√3/4)r² = (√3/4)×196 = 49√3.

  3. Minus

    Segment = 308/3 − 49√3. 90° wala 56 yahan copy mat — 60° pe √3 triangle.

Answer308/3 − 49√3

5. Ring, square leftover, grazed

Annulus / ring: do concentric. Area π(R² − r²) = π(R−r)(R+r).

Ring = bada minus chhota. R=21, r=14 → π(441−196)=770 (22/7).

Example

Outer 21, inner 14, π=22/7. Ring area?

Solution

  1. π(R²−r²)

    22/7 × 245 = 22 × 35 = 770 cm². Trap: π(21−14)²=154.

Answer770 cm²

Square side 14, inscribed circle (diameter 14, r=7): square 196, circle 22/7×49=154, leftover 42.

Horse / rope: peg, rope 14 m, wall se 3/4 circle graze: (3/4)×616=462. (Elevation tower nahi — sirf fraction of disk.)

Combination: figure ko pieces mein kaato (quadrant + rectangle − hole). 3D paint cylinder Surface areas pe — yahan plane figure.

6. Sawal — basic se pro

Q1 · Basic

r=14, π=22/7. Area?

  1. 616 cm²
  2. 88 cm²
  3. 2464 cm²
  4. 44 cm²

Solution

  1. πr²

    616. B circumference. C 4πr² sphere.

Answer616 cm²
Q2 · Basic

r=14, π=22/7. Circumference?

  1. 88 cm
  2. 616 cm
  3. 44 cm
  4. 28 cm

Solution

  1. 2πr

    88. C πr. D diameter.

Answer88 cm
Q3 · Basic

90° sector, r=14. Area?

  1. 154 cm²
  2. 56 cm²
  3. 616 cm²
  4. 22 cm²

Solution

  1. 1/4 × 616

    154. B segment. D arc length.

Answer154 cm²
Q4 · Basic

60° arc, r=21, π=22/7. Length?

  1. 22 cm
  2. 231 cm
  3. 132 cm
  4. 11 cm

Solution

  1. 1/6 × 132

    22. B sector area.

Answer22 cm
Q5 · Medium

r=14, 90°. Minor segment?

  1. 56 cm²
  2. 154 cm²
  3. 98 cm²
  4. 252 cm²

Solution

  1. 154−98

    56. B sector. C triangle. D sector+triangle.

Answer56 cm²
Q6 · Medium

60° sector, r=21. Area?

  1. 231 cm²
  2. 22 cm²
  3. 1386 cm²
  4. 77 cm²

Solution

  1. 1/6 × 1386

    231. C full πr². B arc.

Answer231 cm²
Q7 · Medium

Ring R=21, r=14. Area?

  1. 770 cm²
  2. 154 cm²
  3. 245 cm²
  4. 462 cm²

Solution

  1. 22/7×245

    770. B π(R−r)².

Answer770 cm²
Q8 · Medium

Square side 14, inscribed circle leftover?

  1. 42 cm²
  2. 154 cm²
  3. 196 cm²
  4. 42π cm²

Solution

  1. 196−154

    r=7. 42. B circle khud.

Answer42 cm²
Q9 · Pro

Rope 14 m, 3/4 circle graze, π=22/7. Area?

  1. 462 m²
  2. 616 m²
  3. 154 m²
  4. 88 m²

Solution

  1. 3/4 × 616

    462 m². B poora disk. Tower angle nahi.

Answer462 m²
Q10 · Pro

90° sector r=14 ka perimeter (arc+2r)?

  1. 50 cm
  2. 22 cm
  3. 88 cm
  4. 154 cm

Solution

  1. Arc 22 + 28

    50. B sirf arc. C full C.

Answer50 cm

Practice

Pehle khud, phir neeche kholo.

12345 678910

P1

d=14. Area (π=22/7)?

  1. 154 cm²
  2. 616 cm²
  3. 44 cm²
  4. 88 cm²

Solution

  1. r=7

    22/7×49=154. B r=14 wala.

Answer154 cm²
P2

Semicircle r=14. Area?

  1. 308 cm²
  2. 154 cm²
  3. 88 cm²
  4. 616 cm²

Solution

  1. ½×616

    308. B quadrant / r=7 full.

Answer308 cm²
P3

Major 90° segment r=14?

  1. 560 cm²
  2. 56 cm²
  3. 462 cm²
  4. 154 cm²

Solution

  1. 616−56

    560. B minor.

Answer560 cm²
P4

r=35, 4 rounds, π=22/7. Wire length?

  1. 880
  2. 220
  3. 70
  4. 3850

Solution

  1. 4×220

    C=2πr=220, ×4=880.

Answer880
P5

Sector perimeter r=21, 60°?

  1. 64 cm
  2. 22 cm
  3. 42 cm
  4. 231 cm

Solution

  1. 22+42

    64. B arc only.

Answer64 cm
P6

π(R²−r²) kya hai?

  1. Ring / annulus
  2. Segment
  3. Sphere
  4. Arc

Solution

  1. Concentric

    A. C 4πr².

AnswerA
P7

90° sector r=14. Arc?

  1. 22 cm
  2. 154 cm
  3. 88 cm
  4. 7 cm

Solution

  1. 1/4 × 88

    22.

Answer22 cm
P8

Triangle in 90° sector r=14?

  1. 98 cm²
  2. 56 cm²
  3. 154 cm²
  4. 196 cm²

Solution

  1. ½r²

    98.

Answer98 cm²
P9

θ for quadrant?

  1. 90°
  2. 60°
  3. 180°
  4. 360°

Solution

  1. 1/4

    90°. C semi.

Answer90°
P10

4πr² Class-X is chapter?

  1. Nahi — sphere, Surface areas
  2. Haan, double circle
  3. Arc × 2
  4. Ring

Solution

  1. 3D

    Agla chapter. Yahan πr².

AnswerA

← Index · Constructions

Agle topic: Surface areas and volumes.