Exercise 12.1

Unless stated otherwise, use π =22/7.

1. The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.
Answer:
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.

2. The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.
Answer:
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.

3. Figure. depicts an archery target marked with its five scoring regions from the centre outwards as Gold, Red, Blue, Black and White. The diameter of the region representing Gold score is 21 cm and each of the other bands is 10.5 cm wide. Find the area of each of the five scoring regions.

Answer:

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

4. The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?

Answer:

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

5. Tick the correct answer in the following and justify your choice: If the perimeter and area of a circle are numerically equal, then the radius of the circle is:

(A) 2 units

(B) π units

(C) 4 units

(D) 7 units

Answer:

(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

EXERCISE 12.2

Unless stated otherwise, use π =22/7.

1. Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.
Answer:

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

2. Find the area of a quadrant of a circle whose circumference is 22 cm.

Answer:

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

3. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

Answer:

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

4. A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding : (i) minor segment (ii) major sector. (Use π = 3.14)

Answer:

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

5. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find:
(i) the length of the arc
(ii) area of the sector formed by the arc
(iii) area of the segment formed by the corresponding chord

Answer:

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

6. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. (Use π = 3.14 and √3 = 1.73)

Answer:

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

7. A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. (Use π = 3.14 and √3 = 1.73)

Answer:

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.

8. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see figure). Find:

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

9. A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in figure. Find:

(i) the total length of the silver wire required.

(ii) the area of each sector of the brooch.

Answer:

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

10. An umbrella has 8 ribs which are equally spaced (see figure). Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella.

Answer:

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

11. A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115° . Find the total area cleaned at each sweep of the blades.

Answer:

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

12. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships are warned.
(Use π = 3.14)

Answer:

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

13. A round table cover has six equal designs as shown in figure. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ₹ 0.35 per cm² . (Use √3 = 1.7)

Answer:

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

14. Tick the correct answer in the following :
Area of a sector of angle p (in degrees) of a circle with radius R is
(A) p/180 × 2πR
(B) p/180 × π R²
(C) p/360 × 2πR
(D) p/720 × 2πR²

Answer:
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.

EXERCISE 12.3

Unless stated otherwise, use π =22/7

1. Find the area of the shaded region in figure, if PQ = 24 cm, PR = 7 cm and O is the centre of the circle.

Answer:

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

2. Find the area of the shaded region in figure, if radii of the two concentric circles with centre O are 7 cm and 14 cm respectively and ∠AOC = 40°.

Answer:

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.

3. Find the area of the shaded region in figure, if ABCD is a square of side 14 cm and APD and BPC are semicircles.

Answer:

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.

4. Find the area of the shaded region in figure, where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as centre.

Answer:

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

5. From each corner of a square of side 4 cm a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut as shown in figure. Find the area of the remaining portion of the figure.

Answer:

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

6. In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in figure. Find the area of the design (shaded region).

Answer:

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

7. In figure ABCD is a square of side 14 cm. With centers A, B, C and D, four circles are drawn such that each circle touches externally two of the remaining three circles. Find the area of the shaded region.

Answer:

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

8. Figure depicts a racing track whose left and right ends are semicircular.

The distance between the two inner parallel line segments is 60 m and they are each 106 m long. If the track is 10 m wide, find:

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

(ii) the area of the track.

Answer:

(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

9. In figure, AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.

Answer:

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

10. The area of an equilateral triangle ABC is 17320.5 cm². With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle (see Figure ). Find the area of the shaded region. (Use π = 3.14 and √3 = 1.73205)

Answer:

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

11. On a square handkerchief, nine circular designs each of radius 7 cm are made (see figure). Find the area of the remaining portion of the handkerchief.

Answer:

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

12. In figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the:

(i) quadrant OACB

(ii) shaded region

Answer:

(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

13. In figure, a square OABC is inscribed in a quadrant OPBQ. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14)

Answer:

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

14. AB and CD are respectively arcs of two concentric circles of radii 21 cm and 7 cm and centre O (see Fig. ). If ∠AOB = 30°, find the area of the shaded region.

Answer:

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

15. In figure, ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region.

Answer:

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

16. Calculate the area of the designed region in figure common between the two quadrants of circles of radius 8 cm each.

Answer:

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

EXERCISE 12.3