Class 10 Exam  >  Class 10 Notes  >  Mathematics (Maths) Class 10  >  NCERT Solutions: Areas Related to Circles (Excercise 11.1)

NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

Page No 158


Use π = 22/7 (unless stated otherwise)
Q1: Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.
Sol: Here,
r = 6 cm
θ = 60°
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
∴ Using, the Area of a sector =  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
We have,
Area of the sector with r  = 6 cm and θ = 60°
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

Q2: Find the area of a quadrant of a circle whose circumference is 22 cm.
Sol: Let radius of the circle = r
∴ 2πr = 22
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
⇒  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Here θ = 90°
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

∴ Area of the quadrant  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)of the circle,
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

Q3: The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
Sol: [Length of minute hand] = [radius of the circle]
⇒ r = 14 cm
∵ Angle swept by the minute hand in 60 minutes = 360°
∴ Angle swept by the minute hand in 5 minutes =  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Now, area of the sector with r = 14 cm and θ = 30°
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Thus, the required area swept by the minute hand by 5 minutes = 154/3 cm2.

Q4: 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)
Sol: Given, radius of the circle = 10 cm
Angle subtend by chord at centre = 90° ...(i)
(i) Area of the minor segment = (Area of the sector OAB) - (Area of ΔAOB formed with radius and chord)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
= 3.14 x 25 - 50 = 78.5 - 50 = 28.5 cm2 
(ii) Area of the major segment = Area of the circle - Area of the minor segment
= πr2 - 28.5 = 3.14 x 10 x 10 - 28.5
= 314 - 28.5 = 285.5 cm2

Q5: 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 

Sol:
 NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Here, radius = 21 cm and θ = 60°

(i) Circumference of the circle = 2πr
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

(ii) Area of the sector with sector angle 60°
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

(iii)  Area of the segment formed by the corresponding chord - area of the sector - area of the Δ formed between chord and radius of the circle
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

Q6:  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)
Sol:  

NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
 Radius of the circle = 15 cm
Central angle subtends by chord = 600
Area of sector = NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
= 117.75 cm2

Area of the triangle formed by radii and chord
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

Area of the minor segment = Area of the sector -Area of the triangle formed by radii and chord
= 117.75-97.31 =20.44 cm2 
Area of the circle = πr2 
= 3.14 x 15 x 15 = 706.5 cm2 
Area of the major segment = Area of the circle - Area of the minor segment
= 706.5 - 20.44 = 686.06 cm2

Q7: 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)
Sol:  Here, θ = 120° and r = 12 cm
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
In Δ OAB, ∠O = 120°
⇒∠A + ∠B = 180° − 120 = 60°
∵ OB = OA = 12 cm ⇒∠A = ∠B = 30°
So,  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
⇒  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

In right Δ AMO, 122 − 62 = AM2
⇒ 144 − 36 = AM2
⇒ 108 = AM2
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

Now, from (2),
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
= 36 × 1.73 cm2 = 62.28 cm2     ...(3)
From (1) and (3)
Area of the minor segment = [Area of minor segment] − [Area of Δ AOB]
= [150.72 cm2] − [62.28 cm2] = 88.44 cm2.

Q8:  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:
(i) the area of that part of the field in which the horse can graze.
(ii) the increase in the grazing area if the rope were 10 m long instead of 5 m. (Use π = 3.14)

NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Sol: Here, Length of the rope = 5 m
∴ Radius of the circular region grazed by the horse = 5 m

(i) Area of the circular portion grazed
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)     [∵ θ = 90° for a square field.]
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

(ii) When length of the rope is increased to 10 m,
∴ r = 10 m
⇒ Area of the circular region where θ = 90°.
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
∴ Increase in the grazing area = 78.5 − 19.625 m2 = 58.875 m2.

Page  No 159



Q9:  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 Fig. 
Find: 
(i) the total length of the silver wire required. 
(ii) the area of each sector of the brooch.
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Sol: Diameter of the circle = 35 mm
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
∴  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
(i) Circumference = NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Length of 1 piece of wire used to make diameter to divide the circle into 10 equal sectors = 35 mm
∴ Length of 5 pieces = 5 × 35 = 175 mm
∴ Total length of the silver wire = 110 + 175 mm = 285 mm

(ii) Since the circle is divided into 10 equal sectors,
∴ Sector angle NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
⇒ Area of each sector  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

Q10: 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.
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Sol: Here, radius (r) = 45 cm
Since circle is divided in 8 equal parts,
∴ Sector angle corresponding to each part
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
⇒ Area of a sector (part)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
∴ The required area between the two ribs  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

Q 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.
Sol:  Here, radius (r) = 25 cm
Sector angle (θ) = 115°
∴ Area cleaned by each sweep of the blades
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)   [∵ Each sweep will have to and fro movement]

NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

Q12: 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)
Sol: Here, Radius (r) = 16.5 km
Sector angle (θ) = 80°
∴ Area of the sea surface over which the ships are warned
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

Q13: A round table cover has six equal designs as shown in Fig. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of Rs 0.35 per cm2. (Use √3 = 1.7)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Sol:  Here,    r = 28 cm
Since, the circle is divided into six equal sectors.
∴ Sector angle  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
∴ Area of the sector with θ = 60° and r = 28 cm
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Now, area of 1 design
= Area of segment APB
= Area of sector − Area of ΔAOB  ...(2)
In ΔAOB, ∠AOB = 60°, OA = OB = 28 cm
∴   ∠OAB = 60° and ∠OBA = 60°
⇒ ΔAOB is an equilateral triangle.
⇒ AB = AO = BO
⇒ AB = 28 cm

Draw OM ⊥ AB
∴ In right ΔAOM, we have
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)       ...(3)

Now, from (1), (2) and (3), we have:
Area of segment APQ = 410.67 cm2 − 333.2 cm2 = 77.47 cm2
⇒ Area of 1 design = 77.47 cm2
∴ Area of the 6 equal designs = 6 × (77.47) cm2
= 464.82 cm2
Cost of making the design at the rate of Rs 0.35 per cm2,
= Rs 0.35 × 464.82
= Rs 162.68.

Q14: Tick the correct answer in the following: Area of a sector of angle p (in degrees) of a circle with radius R is
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Sol: Here, radius (r)= R
Angle of sector (θ)= p°
∴ Area of the sector  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)
Thus, the option  NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1) is correct.

The document NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1) is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on NCERT Solutions for Class 10 Maths Chapter 11 - Areas Related to Circles (Excercise 11.1)

1. What is the formula for the area of a circle?
Ans. The formula for the area of a circle is A = πr^2, where A represents the area and r represents the radius of the circle.
2. How do you find the circumference of a circle?
Ans. The circumference of a circle can be found using the formula C = 2πr, where C represents the circumference and r represents the radius of the circle.
3. What is the relation between the radius and diameter of a circle?
Ans. The relation between the radius and diameter of a circle is that the diameter is always twice the length of the radius. In other words, the diameter is equal to 2 times the radius.
4. How do you find the length of an arc in a circle?
Ans. To find the length of an arc in a circle, you can use the formula L = (θ/360) × 2πr, where L represents the length of the arc, θ represents the central angle in degrees, and r represents the radius of the circle.
5. Can the area of a circle be negative?
Ans. No, the area of a circle cannot be negative. The area is always a positive value, representing the amount of space enclosed by the circle's boundary.
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