Mathematics · Textbook solutions

Areas Related to Circles

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 21 questions

11.1 Areas of Sector and Segment of a Circle

21 q

Solved Examples

Worked · 2

11.1 Eg.1 $30^\circ$
11.1 Eg.2 $\angle$

Exercise 11.1

Practice · 19

Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .
Ex 11.1 Q1 $60^\circ$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .
Ex 11.1 Q2 $Find the area of a quadrant of a circle whose circumference is 22 cm.$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .
Ex 11.1 Q3 $The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .
Ex 11.1 Q6 $60^\circ$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .
Ex 11.1 Q7 $120^\circ$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .
Ex 11.1 Q10 $An umbrella has 8 ribs which are equally spaced (see Fig. 11.10). Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella.$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .
Ex 11.1 Q11 $115^\circ$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .
Ex 11.1 Q12 $80^\circ$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .
Ex 11.1 Q13 $^2$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ . A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding : (Use $\pi = 3.14$ )
Ex 11.1 Q4 (i) $minor segment$
Ex 11.1 Q4 (ii) $major sector$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ . In a circle of radius 21 cm, an arc subtends an angle of $60^\circ$ at the centre. Find:
Ex 11.1 Q5 (i) $the length of the arc$
Ex 11.1 Q5 (ii) $area of the sector formed by the arc$
Ex 11.1 Q5 (iii) $area of the segment formed by the corresponding chord$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ . 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 Fig. 11.8). Find (Use $\pi = 3.14$ )
Ex 11.1 Q8 (i) $the area of that part of the field in which the horse can graze.$
Ex 11.1 Q8 (ii) $the increase in the grazing area if the rope were 10 m long instead of 5 m.$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ . 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. 11.9. Find :
Ex 11.1 Q9 (i) $the total length of the silver wire required.$
Ex 11.1 Q9 (ii) $the area of each sector of the brooch.$
Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .
Ex 11.1 Q14 $p$
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What the board has asked from this chapter

Questions per paper · 63 papers · 2023 to 2026

2026

23 q

9 $A circle is divided into 16 identical sectors. If radius of the circle is 7 cm, area of each sector is$
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Area of a Sector
12 $7\,\text{cm}^2$
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Area of a Sector
13 $45^\circ$
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Area of a Sector
13 $\theta$
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Area of a Sector
14 $60^\circ$
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Area of a Sector
14 $\text{cm}^2$
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Area of a Sector
15 $\dfrac{55}{3}\,\text{cm}^2$
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Area of a Sector
15 $The hour hand of a clock is 7 cm long. The angle swept by it between 7:00 a.m. and 8:10 a.m. is :$
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Area of a Sector
15 $60^\circ$
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Area of a Segment
15 $The area of a semicircle of diameter ‘d’ is :$
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Area of a Sector
17 $\theta$
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Area of a Sector
26 $A circle of diameter 20 cm is equally divided into five sectors. Find the area and perimeter of one of the sectors.$
Area of a Sector
27 $\angle AOB = 120^\circ$
Area of a Segment
27 $120^\circ$
Area of a Segment
27 $120^\circ$
Area of a Segment
30 $120^\circ$
Area of a Segment
31 $60^\circ$
Area of a Sector
30 $30^\circ$
Area of a Sector
31 $\theta$
Area of a Sector
A brooch is crafted from silver wire in the shape of a circle with a diameter of 35 cm. The wire is also used to create 5 diameters, dividing the circle into 10 equal sectors as shown in figure. Based on the above information, answer the following questions :
38 (i) $What is the radius of circle ?$
Area of a Sector
A brooch is crafted from silver wire in the shape of a circle with a diameter of 35 cm. The wire is also used to create 5 diameters, dividing the circle into 10 equal sectors as shown in figure. Based on the above information, answer the following questions :
38 (ii) $What is the circumference of the brooch ?$
Area of a Sector
A brooch is crafted from silver wire in the shape of a circle with a diameter of 35 cm. The wire is also used to create 5 diameters, dividing the circle into 10 equal sectors as shown in figure. Based on the above information, answer the following questions :
38 (iii) (a) $What is the total length of silver wire required ?$
Area of a Sector
A brooch is crafted from silver wire in the shape of a circle with a diameter of 35 cm. The wire is also used to create 5 diameters, dividing the circle into 10 equal sectors as shown in figure. Based on the above information, answer the following questions :
38 (iii) (b) $What is the area of each sector of the brooch ?$
Area of a Sector

2025

28 q

1 $The diameter of a wheel is 63 cm. The distance travelled by the wheel in 100 revolutions is :$
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Area of a Sector
3 $144^\circ$
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Area of a Sector
4 $5\pi$
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Area of a Sector
12 $If a large circular pizza is divided into 5 equal sectors, then the central angle of each sector will be :$
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Area of a Sector
13 $54\pi\,\text{cm}^2$
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Area of a Sector
13 $If the length of a chord of a circle is equal to its radius, then the angle subtended by chord at the centre is :$
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Area of a Segment
15 $40\pi$
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Area of a Sector
18 $\dfrac{60}{\pi}$
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Area of a Sector
22 (a) $\angle AOB = 90^\circ$
Area of a Sector
22 (b) $35^\circ$
Area of a Sector
29 $\pi = 3{\cdot}14$
Area of a Segment
29 $The length of the hour hand of a clock is 10 cm. Find the area of the minor sector swept by the hour hand of the clock between 5 a.m. to 8 a.m. Also, find the area of the major sector.$
Area of a Sector
A farmer has a circular piece of land. He wishes to construct his house in the form of largest possible square within the land as shown below. The radius of circular piece of land is 35 m. Based on given information, answer the following questions :
36 (i) $Find the length of wire needed to fence the entire land.$
Area of a Sector
Anurag purchased a farmhouse which is in the form of a semicircle of diameter 70 m. He divides it into three parts by taking a point P on the semicircle in such a way that $\angle PAB = 30^\circ$ as shown in the following figure, where O is the centre of semicircle. In part I, he planted saplings of Mango tree, in part II, he grew tomatoes and in part III, he grew oranges. Based on given information, answer the following questions.
36 (i) $\angle POA$
Area of a Sector
The Olympic symbol comprising five interlocking rings represents the union of the five continents of the world and the meeting of athletes from all over the world at the Olympic games. In order to spread awareness about Olympic games, students of Class-X took part in various activities organised by the school. One such group of students made 5 circular rings in the school lawn with the help of ropes. Each circular ring required 44 m of rope. Also, in the shaded regions as shown in the figure, students made rangoli showcasing various sports and games. It is given that $\Delta OAB$ is an equilateral triangle and all unshaded regions are congruent. Based on above information, answer the following questions :
36 (i) $Find the radius of each circular ring.$
Area of a Sector
A farmer has a circular piece of land. He wishes to construct his house in the form of largest possible square within the land as shown below. The radius of circular piece of land is 35 m. Based on given information, answer the following questions :
36 (ii) $Find the length of each side of the square land on which house will be constructed.$
Area of a Segment
Anurag purchased a farmhouse which is in the form of a semicircle of diameter 70 m. He divides it into three parts by taking a point P on the semicircle in such a way that $\angle PAB = 30^\circ$ as shown in the following figure, where O is the centre of semicircle. In part I, he planted saplings of Mango tree, in part II, he grew tomatoes and in part III, he grew oranges. Based on given information, answer the following questions.
36 (ii) $Find the length of wire needed to fence entire piece of land.$
Area of a Sector
The Olympic symbol comprising five interlocking rings represents the union of the five continents of the world and the meeting of athletes from all over the world at the Olympic games. In order to spread awareness about Olympic games, students of Class-X took part in various activities organised by the school. One such group of students made 5 circular rings in the school lawn with the help of ropes. Each circular ring required 44 m of rope. Also, in the shaded regions as shown in the figure, students made rangoli showcasing various sports and games. It is given that $\Delta OAB$ is an equilateral triangle and all unshaded regions are congruent. Based on above information, answer the following questions :
36 (ii) $\angle AOB$
Area of a Segment
Anurag purchased a farmhouse which is in the form of a semicircle of diameter 70 m. He divides it into three parts by taking a point P on the semicircle in such a way that $\angle PAB = 30^\circ$ as shown in the following figure, where O is the centre of semicircle. In part I, he planted saplings of Mango tree, in part II, he grew tomatoes and in part III, he grew oranges. Based on given information, answer the following questions.
36 (iii) (a) $Find the area of region in which saplings of Mango tree are planted.$
Area of a Segment
A farmer has a circular piece of land. He wishes to construct his house in the form of largest possible square within the land as shown below. The radius of circular piece of land is 35 m. Based on given information, answer the following questions :
36 (iii) (a) $The farmer wishes to grow grass on the shaded region around the house. Find the cost of growing the grass at the rate of ₹ 50 per square metre.$
Area of a Segment
The Olympic symbol comprising five interlocking rings represents the union of the five continents of the world and the meeting of athletes from all over the world at the Olympic games. In order to spread awareness about Olympic games, students of Class-X took part in various activities organised by the school. One such group of students made 5 circular rings in the school lawn with the help of ropes. Each circular ring required 44 m of rope. Also, in the shaded regions as shown in the figure, students made rangoli showcasing various sports and games. It is given that $\Delta OAB$ is an equilateral triangle and all unshaded regions are congruent. Based on above information, answer the following questions :
36 (iii) (a) $R_1$
Area of a Segment
Anurag purchased a farmhouse which is in the form of a semicircle of diameter 70 m. He divides it into three parts by taking a point P on the semicircle in such a way that $\angle PAB = 30^\circ$ as shown in the following figure, where O is the centre of semicircle. In part I, he planted saplings of Mango tree, in part II, he grew tomatoes and in part III, he grew oranges. Based on given information, answer the following questions.
36 (iii) (b) $Find the length of wire needed to fence the region III.$
Area of a Sector
The Olympic symbol comprising five interlocking rings represents the union of the five continents of the world and the meeting of athletes from all over the world at the Olympic games. In order to spread awareness about Olympic games, students of Class-X took part in various activities organised by the school. One such group of students made 5 circular rings in the school lawn with the help of ropes. Each circular ring required 44 m of rope. Also, in the shaded regions as shown in the figure, students made rangoli showcasing various sports and games. It is given that $\Delta OAB$ is an equilateral triangle and all unshaded regions are congruent. Based on above information, answer the following questions :
36 (iii) (b) $Find the length of rope around the unshaded regions.$
Area of a Sector
A farmer has a circular piece of land. He wishes to construct his house in the form of largest possible square within the land as shown below. The radius of circular piece of land is 35 m. Based on given information, answer the following questions :
36 (iii) (b) $Find the ratio of area of land on which house is built to remaining area of circular piece of land.$
Area of a Segment
A brooch is a decorative piece often worn on clothing like jackets, blouses or dresses to add elegance. Made from precious metals and decorated with gemstones, brooches come in many shapes and designs. One such 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 the figure. Based on the above given information, answer the following questions :
37 (i) $Find the central angle of each sector.$
Area of a Sector
A brooch is a decorative piece often worn on clothing like jackets, blouses or dresses to add elegance. Made from precious metals and decorated with gemstones, brooches come in many shapes and designs. One such 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 the figure. Based on the above given information, answer the following questions :
37 (ii) $Find the length of the arc ACB.$
Area of a Sector
A brooch is a decorative piece often worn on clothing like jackets, blouses or dresses to add elegance. Made from precious metals and decorated with gemstones, brooches come in many shapes and designs. One such 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 the figure. Based on the above given information, answer the following questions :
37 (iii) (a) $Find the area of each sector of the brooch.$
Area of a Sector
A brooch is a decorative piece often worn on clothing like jackets, blouses or dresses to add elegance. Made from precious metals and decorated with gemstones, brooches come in many shapes and designs. One such 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 the figure. Based on the above given information, answer the following questions :
37 (iii) (b) $Find the total length of the silver wire used.$
Area of a Sector

2024

21 q

4 $\angle C = \angle D = 90^\circ$
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Area of a Sector
5 $5\sqrt{2}$
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Area of a Segment
5 $90^\circ$
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Area of a Sector
6 $60\pi$
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Area of a Sector
6 $\dfrac{7}{20}$
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Area of a Sector
8 $60^\circ$
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Area of a Sector
9 $10\pi$
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Area of a Sector
12 $A chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is :$
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Area of a Segment
17 $90^\circ$
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Area of a Sector
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct answer to these questions from the codes (A), (B), (C) and (D) as given below.
20 $= 2\pi \times$
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Area of a Sector
22 (a) $The minute hand of a clock is 14 cm long. Find the area on the face of the clock described by the minute hand in 5 minutes.$
Area of a Sector
22 (b) $60^\circ$
Area of a Sector
27 $150^\circ$
Area of a Sector
31 $\pi = 3{\cdot}14$
Area of a Sector
32 $60^\circ$
Area of a Segment
35 $The perimeter of a certain sector of a circle of radius 5.6 m is 20.0 m. Find the area of the sector.$
Area of a Sector
35 $\angle AOB = 60^\circ$
Area of a Segment
A stable owner has four horses. He usually tie these horses with 7 m long rope to pegs at each corner of a square shaped grass field of 20 m length, to graze in his farm. But tying with rope sometimes results in injuries to his horses, so he decided to build fence around the area so that each horse can graze. Based on the above, answer the following questions :
36 (i) $Find the area of the square shaped grass field.$
Area of a Sector
A stable owner has four horses. He usually tie these horses with 7 m long rope to pegs at each corner of a square shaped grass field of 20 m length, to graze in his farm. But tying with rope sometimes results in injuries to his horses, so he decided to build fence around the area so that each horse can graze. Based on the above, answer the following questions :
36 (ii) (a) $Find the area of the total field in which these horses can graze.$
Area of a Sector
A stable owner has four horses. He usually tie these horses with 7 m long rope to pegs at each corner of a square shaped grass field of 20 m length, to graze in his farm. But tying with rope sometimes results in injuries to his horses, so he decided to build fence around the area so that each horse can graze. Based on the above, answer the following questions :
36 (ii) (b) $\pi = 3.14$
Area of a Sector
A stable owner has four horses. He usually tie these horses with 7 m long rope to pegs at each corner of a square shaped grass field of 20 m length, to graze in his farm. But tying with rope sometimes results in injuries to his horses, so he decided to build fence around the area so that each horse can graze. Based on the above, answer the following questions :
36 (iii) $What is area of the field that is left ungrazed, if the length of the rope of each horse is 7 cm ?$
Area of a Sector

2023

19 q

2 $90^\circ$
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Area of a Sector
5 $What is the area of a semi-circle of diameter 'd' ?$
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Area of a Sector
5 $4 : 5$
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Area of a Sector
13 $The hour-hand of a clock is 6 cm long. The angle swept by it between 7:20 a.m. and 7:55 a.m. is :$
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Area of a Sector
26 $In the given figure, AB and CD are diameters of a circle with centre O perpendicular to each other. If OA = 7 cm, find the area of shaded region.$
Area of a Segment
27 $60^\circ$
Area of a Segment
29 $120^\circ$
Area of a Sector
31 $60^\circ$
Area of a Sector
31 $\pi = 3{\cdot}14$
Area of a Segment
34 $60^\circ$
Area of a Segment
35 $\pi = 3{\cdot}14$
Area of a Sector
In an annual day function of a school, the organizers wanted to give a cash prize along with a memento to their best students. Each memento is made as shown in the figure and its base ABCD is shown from the front side. The rate of silver plating is ₹ 20 per $\text{cm}^2$ . Based on the above, answer the following questions :
36 (i) $What is the area of the quadrant ODCO ?$
Area of a Sector
In an annual day function of a school, the organizers wanted to give a cash prize along with a memento to their best students. Each memento is made as shown in the figure and its base ABCD is shown from the front side. The rate of silver plating is ₹ 20 per $\text{cm}^2$ . Based on the above, answer the following questions :
36 (ii) $\Delta AOB$
Area of a Sector
In an annual day function of a school, the organizers wanted to give a cash prize along with a memento to their best students. Each memento is made as shown in the figure and its base ABCD is shown from the front side. The rate of silver plating is ₹ 20 per $\text{cm}^2$ . Based on the above, answer the following questions :
36 (iii) (a) $What is the total cost of silver plating the shaded part ABCD ?$
Area of a Sector
In an annual day function of a school, the organizers wanted to give a cash prize along with a memento to their best students. Each memento is made as shown in the figure and its base ABCD is shown from the front side. The rate of silver plating is ₹ 20 per $\text{cm}^2$ . Based on the above, answer the following questions :
36 (iii) (b) $What is the length of arc CD ?$
Area of a Sector
Governing council of a local public development authority of Dehradun decided to build an adventurous playground on the top of a hill, which will have adequate space for parking. After survey, it was decided to build rectangular playground, with a semi-circular area allotted for parking at one end of the playground. The length and breadth of the rectangular playground are 14 units and 7 units, respectively. There are two quadrants of radius 2 units on one side for special seats. Based on the above information, answer the following questions :
38 (i) $What is the total perimeter of the parking area ?$
Area of a Sector
Governing council of a local public development authority of Dehradun decided to build an adventurous playground on the top of a hill, which will have adequate space for parking. After survey, it was decided to build rectangular playground, with a semi-circular area allotted for parking at one end of the playground. The length and breadth of the rectangular playground are 14 units and 7 units, respectively. There are two quadrants of radius 2 units on one side for special seats. Based on the above information, answer the following questions :
38 (ii) (a) $What is the total area of parking and the two quadrants ?$
Area of a Sector
Governing council of a local public development authority of Dehradun decided to build an adventurous playground on the top of a hill, which will have adequate space for parking. After survey, it was decided to build rectangular playground, with a semi-circular area allotted for parking at one end of the playground. The length and breadth of the rectangular playground are 14 units and 7 units, respectively. There are two quadrants of radius 2 units on one side for special seats. Based on the above information, answer the following questions :
38 (ii) (b) $What is the ratio of area of playground to the area of parking area ?$
Area of a Sector
Governing council of a local public development authority of Dehradun decided to build an adventurous playground on the top of a hill, which will have adequate space for parking. After survey, it was decided to build rectangular playground, with a semi-circular area allotted for parking at one end of the playground. The length and breadth of the rectangular playground are 14 units and 7 units, respectively. There are two quadrants of radius 2 units on one side for special seats. Based on the above information, answer the following questions :
38 (iii) $Find the cost of fencing the playground and parking area at the rate of ₹ 2 per unit.$
Area of a Sector

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 3 times: 2023*, 2024*, 2026 (* = numbers or wording changed that year)
14 $60^\circ$
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Question 2
Asked 2 times: 2023, 2026* (* = numbers or wording changed that year)
31 $60^\circ$
Question 3
Asked 2 times: 2023, 2026
15 $The area of a semicircle of diameter ‘d’ is :$
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Question 4 (case study, 4 parts)
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
A brooch is crafted from silver wire in the shape of a circle with a diameter of 35 cm. The wire is also used to create 5 diameters, dividing the circle into 10 equal sectors as shown in figure. Based on the above information, answer the following questions :
38 (i) $What is the radius of circle ?$
38 (ii) $What is the circumference of the brooch ?$
38 (iii) (a) $What is the total length of silver wire required ?$
38 (iii) (b) $What is the area of each sector of the brooch ?$
Question 5
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
15 $The hour hand of a clock is 7 cm long. The angle swept by it between 7:00 a.m. and 8:10 a.m. is :$
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Question 6
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
31 $\theta$
Question 7 (case study, 4 parts)
Asked 2 times: 2023*, 2024 (* = numbers or wording changed that year)
A stable owner has four horses. He usually tie these horses with 7 m long rope to pegs at each corner of a square shaped grass field of 20 m length, to graze in his farm. But tying with rope sometimes results in injuries to his horses, so he decided to build fence around the area so that each horse can graze. Based on the above, answer the following questions :
36 (i) $Find the area of the square shaped grass field.$
36 (ii) (a) $Find the area of the total field in which these horses can graze.$
36 (ii) (b) $\pi = 3.14$
36 (iii) $What is area of the field that is left ungrazed, if the length of the rope of each horse is 7 cm ?$
Question 8
This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2025
29 $The length of the hour hand of a clock is 10 cm. Find the area of the minor sector swept by the hour hand of the clock between 5 a.m. to 8 a.m. Also, find the area of the major sector.$
Question 9
This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026
13 $45^\circ$
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Question 10
This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026
27 $\angle AOB = 120^\circ$

11.1 Areas of Sector and Segment of a Circle

Solved Examples

Exercise 11.1

What the board has asked from this chapter

2026

2025

2024

2023

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4 (case study, 4 parts)

Question 5

Question 6

Question 7 (case study, 4 parts)

Question 8

Question 9

Question 10