NCERT Solutions for Class 10 Maths Chapter 12: Areas Related to Circles

Let the radius of the third circle be R.
Circumference of the circle with radius R = 2πR
Circumference of the circle with radius 19 cm = 2π × 19 = 38π cm
Circumference of the circle with radius 9 cm = 2π × 9 = 18π cm
Sum of the circumference of two circles = 38π + 18π = 56π cm
Circumference of the third circle = 2πR = 56π
⇒ 2πR = 56π cm
⇒ R = 28 cm
The radius of the circle which has circumference equal to the sum of the circumferences of the two circles is 28 cm.

Let the radius of the third circle be R.
Area of the circle with radius R = πR2
Area of the circle with radius 8 cm = π × 82 = 64π cm2
Area of the circle with radius 6 cm = π × 62 = 36π cm2
Sum of the area of two circles = 64π cm2 + 36π cm2 = 100π cm2
Area of the third circle = πR2 = 100π cm2
⇒ πR2 = 100π cm2
⇒ R2 = 100 cm2
⇒ R = 10 cm
Thus, the radius of the new circle is 10 cm.

Diameter of Gold region= 21 cm
Radius of gold region= 21/2= 10.5 cm


Area of Gold Region = πr²
= π(10.5)² =( 22/7)×  110.25 = 346.5 cm²
Area of Gold Region= 345.5 cm²

Radius of red region = Radius for gold + red region= 10.5 + 10.5= 21 cm
Area of Red Region = π²(21² - 10.5²)

[Area of a ring= π (R²-r²), where R= radius of outer ring & r= radius of inner ring]

= 22/7 (21² – 10.5²)        [ a²-b²= (a+b)(a-b)]
= 22/7 (21 + 10.5)(21 – 10.5)
= (22/7 )x 31.5 x 10.5 = 1039.5 cm²

Area of Red Region = 1039.5 cm²


Radius of blue region = Radius of blue region = Now radius for gold + red+ blue region= 21+10.5= 31.5 cm

Area of Blue Region = π(31.5² – 21²)
= 22/7 (31.5² - 21²)
= 22/7 (31.5 +21)(31.5 - 21)
= (22/7 )x 52.5 x 10.5 = 1732.5 cm²

Area of Blue Region =1732.5 cm²


Now,
Radius of black region= radius for gold + red+ blue + black region= 31.5+10.5= 42 cm

Area of Black Region = π(42² – 31.5²)

= 22/7 (42²-31.5² )
= 22/7 (42+31.5)(42-31.5)
= (22/7 )x 73.5 x 10.5 = 2425.5 cm²

Area of Black Region =2425.5 cm²

Now
Radius of white region= radius for gold + red+ blue + black+ white region= 42+10.5= 52.5 cm

Area of White Region= π(52.5² – 42²)
= 22/7 (52.5²-42² )
= 22/7 (52.5+42)(52.5-42)
= (22/7 )x 94.5 x 10.5 = 3118.5 cm²

Area of white Region =3118.5 cm²

The car travels in 10 minutes =66 / 6
= 11km
= 1100000cm
Circumference of the wheel = distance covered by the wheel in one revolution Thus, we have,
Circumference = 2 × 22/7 × 80/2 = 251.43cm
Thus, the number of revolutions covered by the wheel in 1100000cm = 1100000 / 251.43 ≈ 4375

(A) Circumference = Area

Correct option is A)
Perimeter of circle  = 2πr
Area of circle  = πr²
According to the Question,
Perimeter of circle = Area of circle
2πr = πr²
or, 2 = r
or, r = 2 units

We know that in 1 hour (i.e., 60 minutes), the minute hand rotates 360°
In 5 minutes, minute hand will rotate = 360°/ 60 x 5 = 30°
Therefore, the area swept by the minute hand in 5 minutes will be the area of a sector of 30° in a circle of 14 cm radius.
Area of sector of angle θ = θ/360° . π. r2
Area of sector of 30° = 30°/360° . 22/7 . 14 .14
→ = 22/12 . 2 .14

= ((11).(14))/3
= 154/3 cm2
= 51.33 cm2

Therefore, the area swept by the minute hand in 5 minutes is 51.33 cm2

Area of the sector making angle θ
= (θ/360°)×π r2
Area of the sector making angle 60°
= (60°/360°)×π r2 cm2
= (1/6)×62π 
= 36/6 π cm2
= 6 × 22/7 cm2
= 132/7 cm2

Quadrant of a circle means sector is making angle 90°.
Circumference of the circle = 2πr
= 22 cm
Radius of the circle = r
= 22/2π cm
= 7/2 cm
Area of the sector making angle 90°
= (90°/360°)×π r2 cm2
= (1/4)×(7/2)2π 
= (49/16) π cm2
= (49/16) × (22/7) cm2
= 77/8 cm2

Radius of the circle = 10 cm


Major sector is making 360° − 90° = 270°


Area of the sector making angle 270°


=(270°/360°) × πr2 cm2


=(3/4) × 102π = 75πcm2


=75 × 3.14cm2 = 235.5cm2

∴ Area of the major sector = 235.5cm2


Height of ΔAOB = OA = 10 cm


Base of ΔAOB = OB = 10 cm


Area of ΔAOB = 1/2 × OA × OB


= 1/2 × 10 × 10 = 50cm2


Minor sector is making 90°


Area of the sector making angle 90°


= (90°/360°) × πr2 cm2


= (1/4) × 102π = 25πcm2


= 25 × 3.14 cm2

= 78.5 cm2


Area of the minor segment = Area of the sector making angle 90° − Area of ΔAOB


= 78.5 cm2 − 50cm2 = 28.5cm2

In the mentioned figure,
O is the centre of circle,
AB is a chord
AXB is a major arc,
OA=OB= radius =21 cm
Arc AXB subtends an angle 60° at O.

i) Length of an arc AXB = 60/360 × 2π × r

= 1/6 × 2 × 22/7 × 21

= 22cm

ii) Area of sector AOB = 60/360 × π × r2

= 1/6 × 22/7 × (21)2

= 231cm2

(iii) Area of equilateral ΔAOB = √3/4 × (OA)2 = √3/4 × 212 = (441√3)/4 cm2

Area of the segment formed by the corresponding chord
= Area of the sector formed by the arc - Area of equilateral ΔAOB
= 231 cm2 - (441√3)/4 cm2

Radius of the circle = 15 cm
ΔAOB is isosceles as two sides are equal.
∴ ∠A = ∠B
Sum of all angles of triangle = 180°
∠A + ∠B + ∠C = 180°
⇒ 2 ∠A = 180° - 60°
⇒ ∠A = 120°/2
⇒ ∠A = 60°
Triangle is equilateral as ∠A = ∠B = ∠C = 60°
∴ OA = OB = AB = 15 cm
Area of equilateral ΔAOB
= √3/4 × (OA)2 = √3/4 × 152
= (225√3)/4 cm2 = 97.3 cm2

Angle subtend at the centre by minor segment = 60°
Area of Minor sector making angle 60°
= (60°/360°) × π r2 cm2
= (1/6) × 152 π  cm2 =  225/6 π  cm2
=  (225/6) × 3.14 cm2 = 117.75  cm2

Area of the minor segment = Area of Minor sector - Area of equilateral ΔAOB
= 117.75  cm2 - 97.3 cm2 = 20.4 cm2

Angle made by Major sector = 360° - 60° = 300°
Area of the sector making angle 300°
= (300°/360°) × π r2 cm2
= (5/6) × 152 π  cm2 =  1125/6 π  cm2
=  (1125/6) × 3.14 cm2 = 588.75  cm2

Area of major segment = Area of Minor sector + Area of equilateral ΔAOB
= 588.75  cm2 + 97.3 cm2 = 686.05 cm2

We have,
In a circle,
Radius
r = 12cm.
θ = 120°
Area of segment APB = Area of sector OAPB - Area of ΔOAB
Area of sector OAPB
= θ/360° × πr2
= 120°/360 × 3.14 × (12)2
= 1/3 × 3.14 × 144
= 3.14 × 4 ×12
= 3.14 × 48
= 150.72 cm2
Finding area of triangle ΔAOB
Area of ΔAOB = 12 × Base × Height
We draw
OM⊥AB
∴ ∠OMB = ∠OMA = 90°
In
ΔOMA and ΔOMB
∠OMA = ∠OMB (Both 90°)
OM = OM( Both radius)
OA = OB (Common line)
∴ ΔOMA ≅ ΔOMB (S.A.S. Congurency)
⇒ ∠AOM = ∠BOM
Then,
BM = AM = 1/2 AB ....... (1)
Now,
∠AOM = ∠BOM = 1/2 ∠BOA
= 1/2 × 1200
= 60°

In right angle triangle OMA
sin O= AM/AO
sin 60° = AM/12
√3/2 = AM/12
AM = 6√3
Again, In right angle triangle OMA
cos O = OM/AO
cos 60° = OM/AO
1/2 = OM/12
OM = 6cm.
Now, from equation (1) and we get,
AM = 1/2 AB
2AM = AB
AB = 2AM
AB = 2 × 6√3  ∴ AM = 6√3
AB = 12√3 cm.
Now,
Area of ΔAOB = 1/2 × Base × Height
= 1/2 × AB × OM
= 1/2 × 12√3 × 6
= 36√3
= 36 × 1.73
62.28 cm2
Hence, the Area of segment of
APB = Area of sector OAPB − Area of ΔOAB
= 150.72 − 62.28
= 88.44 cm2
Hence, this is the answer.

Side of square field = 15 m
Length of rope is the radius of the circle, r = 5 m
Since, the horse is tied at one end of square field; it will graze only quarter of the field with radius 5 m.
(i) Area of circle = π r2 = 3.14 × 52 = 78.5 m2
Area of that part of the field in which the horse can graze = 1/4 of area of the circle

= 78.54 = 19.625 m2

Diameter =35mm


∴ Circumference of circle
= 2 × π × r = 2 × π × 17.5 = 110 mm

i) The total length of the silver wire required


= Circumference of circle + (5 × 35)


= 110 + 175 = 285mm


(ii) The area of each sector of the brooch


= Area of circle/10


= (π ×17.5 × 17.5)/10 
 = 385/4 mm2

Here, area between two consecutive ribs = 1/8 × Area of umbrella
Radius = r = 45cm
Area of umbrella = πr2
= 22/7 × 45 × 45 cm2
= 22/7 × 2025 cm2

Required area = 1/8 × Area of umbrella
= 1/8 × 22/7 × 2025 cm2
= 22275/28 cm2
= 795.54 cm2
Hence, area between two consecutive ribs is 795.54 cm2

Angle of the sector of circle made by wiper = 115°
Radius of wiper = 25 cm
Area of the sector made by wiper
= (115°/360°) × π r2 cm2
= 23/72 × 22/7 × 252
=  23/72 × 22/7 × 625 cm2
= 158125/252 cm2

Total area cleaned at each sweep of the blades
= 2 ×158125/252 cm2
=  158125/126 = 1254.96 cm2

Let O bet the position of Lighthouse.
Distance over which light spread i.e. radius, r = 16.5 km
Angle made by the sector = 80°
Area of the sea over which the ships are warned = Area made by the sector.
Area of sector = (80°/360°) × π r2 km2
= 2/9 × 3.14 × (16.5)2 km2
=  189.97 km2

r = 28 cm, θ = 360°/6 = 60°

Area OAMB = θ/360° × πr2 = 60°/360° × 22/7 × 28 × 28

= 1232/3 cm2 = 410.67 cm2

Now, in ΔONA and ΔONB
i) ∠ONA = ∠ONB [each 90°]
ii) OA = OB [Radii of common circle]
iii) ON = ON [common]

∴ ΔONA ≅ ΔONB [By RHS congruency]
Hence, AN = NB = 1/2 AB

& ∠AON = ∠BON = 1/2 ∠AOB = 60°/2 = 30°

Now in ΔONA, cos 30° = ON/OA ⇒ √3/2 = ON/28 ⇒ ON = 14√3cm

& sin 30° = AN/OA ⇒ 1/2 = AN/28 ⇒ AN = 14cm

& 2AN = 14 × 2 = 28cm = AB

∴  arΔAOB = 1/2 × AB × ON = 1/2 × 28 × 14√3 = 196√3 = 196 × 1.7 = 333.2 cm2

∴ Area of one design = 410.67 − 333.2 = 77.47 cm2

∴ Area of six design = 6 × 77.47 = 464.82 cm2

Therefore, Cost of making the designs = Rs. (464.82 × 0.35)

= Rs.162.68

Area of a sector of angle p
= p/360 × π R2
= p/360 × 2/2 × π R2
=  2p/720 × 2πR2
Hence, Option (D) is correct.

Side of square = 14 cm
Four quadrants are included in the four sides of the square.
∴ Radius of the circles = 14/2 cm = 7 cm
Area of the square ABCD = 142 = 196 cm2
Area of the quadrant = (π R2)/4 cm2 = (22/7 × 72)/4 cm2
= 77/2 cm2
Total area of the quadrant = 4 × 77/2 cm2 = 154 cm2

Area of the shaded region = Area of the square ABCD - Area of the quadrant
= 196 cm2 - 154 cm2
= 42 cm2

PQ = 24 cm and PR = 7 cm


∠P = 90° (Angle in the semi-circle)


∴ QR is hypotenuse of the circle = Diameter of the circle.


By Pythagoras theorem,


QR2 = PR2 + PQ2


⇒ QR2 = 72 + 242

⇒ QR2 = 49 +576

⇒ QR2 = 625

⇒ QR = 25cm


∴ Radius of the circle = 25/2cm


Area of the semicircle = (πR2)/2 


= (22/7 × 25/2 × 25/2)/2) cm2

= 13750/56 cm2 = 245.54 cm2


Area of the ΔPQR = 1/2 × PR × PQ


= 1/2 × 7 × 24 cm2

= 84 cm2


Area of the shaded region


= Area of the semicircle - Area of the ΔPQR


= 245.54 cm2 − 84 cm2 = 161.54cm2

Given that,
Radius of the small circle, OB = 7 cm
Radius of second circle, OA = 14 cm
and ∠AOC = 40°

We know that, the area of a sector that subtends an angle θ at the centre of the circle is θ/360° × πr2
where, θ is in degrees.

∴ Area of minor sector OBD = 40/360 × 22/7 × 7 × 7 [∵ π = 22/7]

= 1/9 × 22 × 7

= 17.11 cm2

Also, area of minor sector OAC = 40/360 × 22/7 × 14 × 14

= 1/9 × 22 × 2 × 14

= 68.4 cm2

Now, area of the shaded region = Area of sector OAC − Area of sector OBD

= 68.4 − 17.1

= 51.3 cm2

Hence, the area of the shaded region is 51.3 cm2.

Area of the shaded region = Area of Square ABCD − (Area of semicircle APD + Area of semicircle BPC)

= (14)2 − [1/2 π (14/2)2 + 1/2 π (14/2)2]

= (14)2 − 22/7 (7)2

= 196 − 154

= 42 cm2

Hence, the area of the shaded region is 42 cm2.

Area of Shaded region = Area of equilateral triangle ABO + Area of Major sector


Area of Equilateral Triangle ABO = √3/4 × a2 = √3/4 × 12 × 12 cm2



= 62.352 cm2


Area of major sector = θ/360 × π × r2

= 300/360 × π × 6 × 6 cm2
= 94.2 cm2

Area of Shaded region = 62.352 + 94.2 = 156.55 cm2

Side of the square = 4 cm
Radius of the circle = 1 cm
Four quadrant of a circle are cut from corner and one circle of radius are cut from middle.
Area of square = (side)2 = 42 = 16 cm2
Area of the quadrant = (π R2)/4 cm2 = (22/7 × 12)/4 = 11/14 cm2
∴ Total area of the 4 quadrants = 4 × (11/14) cm2 = 22/7 cm2
Area of the circle = π R2 cm2 = (22/7 × 12) = 22/7 cm2
Area of the shaded region = Area of square - (Area of the 4 quadrants + Area of the circle)
= 16 cm2 - (22/7 + 22/7) cm2
= 68/7 cm2

cos θ = AD/OA

cos 30° = √3/2

AD = √3/2 × 32

AD = 16√3cm

AB = 32√3cm

shaded region = Area of circle - Area of triangle

= π(32)2 − 3 × 1/2 × 32√3 × 16

= ((22578/7) − 768√3)cm2

(i) 
The distance around the track along its inner edge


= EG + FH + 2 × (circumference of the semicircle of radius OE = 30 cm)


= 106 + 106 + 2(1/2 × 2π × 30) = 212 + 60π


= 212 + 60 x 22/7 = (212+1320/7)m = (1484 + 1320)/7m = 2804/7m = (400)4/7m


(ii) 
Area of the track = Area of the shaded region


= Area of the rectangle AEGC + Area of the rectangle BFHD + 2 + (Area of the semicircle of radius 40 m − Area of the semicircle with radius 30 m−)


= [(10 × 106) + (10 × 106)] + 2{1/2 × 22/7 × (40)2 − 1/2 × 22/7 × (30)2}


= 1060 + 1060 + 22/7 [(40)2 − (30)2]


= 2120 + 22/7 × 700 = 2120 + 2200 = 4320 m2

OD is diameter of smaller circle.



Area of the smaller circle


= π × 3.52 = 38.49 cm2



Area of ΔABC = 0.5 × (14)(7) = 49 cm2



Area of semi-circle = 0.5π(7)(7) = 76.97 cm2



Area of shaded portion in semi-circle = 76.97 − 49 = 27.97 cm2



Total required area = 38.49 + 27.97 = 66.46 cm2

Let the side of the equilateral triangle be a.
Area of equilateral triangle = 17320.5 cm2
√3/4 (a)2 = 17320.5

1.73205/4 a2 = 17320.5

a2 = 4 x 10000

a = 200 cm

Each sector is of measure 60°.
Area of sector ADEF = 60°/360° x π x r2

= 1/6 x π x (100)2

= (3.14 x 10000)/6

= 15700/3 cm2

Area of shaded region = Area of equilateral triangle − 3 × Area of each sector

= 17320.5 - 3 x 15700/3
= 17320.5 - 15700 = 1620.5 cm2

Side of square = 3 × diameter of 1 circle
= 3 × 14 = 42cm

Area of remaining portion = Area of square - 9 × area of 1 circle

= (42 × 42) - 9 × 22/7 × 7 × 7

= 1764 − 1386

= 378 cm2

(i) Area of quadrant OACB = Area of Circle/4 = (π × r2)/4 = 1/4 × 22/7 × 3.5 × 3.5

= 9.625cm2

(ii) Area of the shaded region = Area of Quadrant − Area of △BDO

= 9.625 − (1/2 × 3.5 × 2)

= 6.125 cm2

Side of square = OA = AB = 20 cm
Radius of the quadrant = OB
OAB is right angled triangle
By Pythagoras theorem in ΔOAB ,
OB2 = AB2  + OA2
⇒ OB2 = 202  + 202
⇒ OB2 = 400  + 400
⇒ OB2 = 800
⇒ OB = 20√2 cm
Area of the quadrant = (πR2)/4 cm2 = 3.14/4 × (20√2)2 cm2 = 628 cm2
Area of the square = 20 × 20 = 400  cm2

Area of the shaded region = Area of the quadrant - Area of the square
= 628 - 400 cm2 = 228 cm2

Radius of the larger sector, R=21 cm


Radius of the smaller sector, r=7 cm


Angle subtended by sectors of both concentric circles = 30°



Area of the sector making angle θ


= (θ/360°) × πr2



Area of the larger sector

= (30°/360°) × πR2cm2
= 1/12 × 22/7 × 212 cm2
= 1/12 × 22/7 × 21 × 21 cm2
= 231/2 cm2



Area of the smaller circle

= (30°/360°) × πr2 cm2
= 1/12 × 22/7 × 72 cm2
= 1/12 × 22/7 × 7 × 7 cm2
= 77/6 cm2



Area of the shaded region = Area of the larger sector − Area of the smaller sector


= (231/2 − 77/6) cm2
= 616/6 cm2
= 308/3 cm2
= (102) 2/3 cm2

Radius of the the quadrant ABC of circle = 14 cm
AB = AC = 14 cm
BC is diameter of semicircle.
ABC is right angled triangle.
By Pythagoras theorem in ΔABC,
BC2 = AB2  + AC2
⇒ BC2 = 142  + 142
⇒ BC = 14√2 cm
Radius of semicircle = 14√2/2 cm = 7√2 cm

Area of ΔABC = 1/2 × 14 × 14 = 98 cm2
Area of quadrant = 1/4 × 22/7 × 14 × 14 = 154 cm2
Area of the semicircle = 1/2 × 22/7 × 7√2 × 7√2 = 154 cm2

Area of the shaded region = Area of the semicircle + Area of ΔABC - Area of quadrant
= 154 + 98 - 154 cm2 = 98 cm2

AB = BC = CD = AD = 8 cm

Area of ΔABC = Area of ΔADC = 1/2 × 8 × 8 = 32 cm2
Area of quadrant AECB = Area of quadrant AFCD = 1/4 × 22/7 × 82
= 352/7 cm2

Area of shaded region = (Area of quadrant AECB - Area of ΔABC) +(Area of quadrant AFCD - Area of ΔADC)
= (352/7 - 32) + (352/7 -32) cm2
= 2 × (352/7 -32) cm2
=  256/7 cm2