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From the wheels we ride to the plates we use, areas of circles are everywhere, and math shows us how to measure them. Have you ever wondered how one determines the amount of material required for a round tablecloth, or how engineers calculate the area of a circular stadium? Areas related to circles are used to solve problems in real life involving circles. As per the latest NCERT syllabus, this chapter contains the basic concepts of calculating the circumference, area, and components of a circle, such as sectors and segments. They find extensive usage in fields such as architecture, sports, and engineering, and therefore are extremely practical to use in everyday life. These NCERT solutions for class 10 prove that even endless curves are measurable with a finite formula.
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Defining the area of a circle means capturing the space within its boundary, turning curves into clear measures. The NCERT solutions of class 10 Maths Chapter 11 Areas related to circles, are designed by our experienced subject experts to offer a systematic, structured approach and step-by-step solutions to these important concepts and help students to prepare well for exams and to gain knowledge about all the natural processes happening around them by a series of solved questions given in the NCERT textbook exercise. It covers questions from all the topics and will help you improve your speed and accuracy. Get syllabus details, key notes, and downloadable PDFs directly from the following link: NCERT
Class 10 Maths chapter 11 solutions Exercise: 11.1 Page number: 158-159 Total questions: 14 |
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Q1: Find the area of a sector of a circle with radius 6 cm if the angle of the sector is 60°.
Answer:
We know that the area of a sector having radius r and angle Θ is given by:-
Thus, the area of the given sector is:-
=
=
Q2: Find the area of a quadrant of a circle whose circumference is 22 cm.
Answer:
We are given the circumference of the circle.
Thus,
2πr = 22
Also, we know that the area of a sector is given by :
It is given that we need to find the area of a quadrant, thus Θ = 90∘
Hence, the area becomes:-
=
=
Answer:
The minute hand rotates 360° in one hour.
We need to find a rotation in 5 minutes. :-
=
The area of the sector is given by :
Hence, the area swept by the minute hand in 5 minutes is
Q4: A chord of a circle of radius 10 cm subtends a right angle at the center.
Find the area of the corresponding : (i) minor segment
Answer:
The angle in the minor sector is 90o.
Thus, the area of the sector is given by:-
Now the area of a triangle is:-
Thus, the area of the minor segment
= Area of the sector - Area of a triangle
= 78.5 − 50 = 28.5 cm2
Q4: A chord of a circle of radius 10 cm subtends a right angle at the center.
Find the area of the corresponding : (ii) major sector. (Use π = 3.14)
Answer:
The area of the major sector can be found directly by using the formula :
In this case, the angle is 360o - 90o = 270°.
Thus, the area is: -
=
=
Q5: In a circle of radius 21 cm, an arc subtends an angle of 60° at the center.
Find: (i) the length of the arc
Answer:
The length of the arc is given by:-
Length of arc =
=
Hence, the length of the arc is 22 cm.
Q5: In a circle of radius 21 cm, an arc subtends an angle of 60° at the center.
Find: (ii) the area of the sector formed by the arc
Answer:
We know that the area of the sector is given by:-
=
Thus, the area of the sector is 231 cm2.
Q5: In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre.
Find: (iii) area of the segment formed by the corresponding chord
Answer:
For the area of the segment, we need to subtract the area of the triangle attached from the area of the arc.
Thus, consider the triangle:-
It is given that the angle of arc is 60o, or we can say that all angles are 60o (since two sides are equal). Hence, it is an equilateral triangle.
Area of the triangle is:-
=
=
=
Hence, the area of the segment is:-
=
Answer:
The area of the sector is :
= 117.85 cm2
Now consider the triangle; the angle of the sector is 600.
This implies it is an equilateral triangle. (As two sides are equal, so will have the same angle. This is possible only when all angles are equal, i.e., 60o .)
Thus, the area of the triangle is:-
=
Hence area of the minor segment: =
And the area of the major segment is :
=
Answer:
For the area of the segment, we need the area of the sector and the area of the associated triangle.
So, the area of the sector is :
=
=
Now, consider the triangle:-
Draw a perpendicular from the centre of the circle to the base of the triangle (let it be h).
Using geometry, we can write,
or h = 6 cm
Similarly,
or
Thus, the area of the triangle is :
=
= 62.28 cm2
Hence, the area of segment is: = 150.72 − 62.28 = 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 Fig.). Find (i) the area of that part of the field in which the horse can graze.
Answer:
The part grazed by the horse is given by = Area of the sector
=
= 19.62 m2
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 Fig.). Find (ii) the increase in the grazing area if the rope were 10 m long instead of 5 m. (Use π=3.14 )
Answer:
When the length of the rope is 10 m, the area grazed will be:-
= 25π m2
Hence, the change in the grazing area is given by :
=
Q9: A brooch is made with silver wire in the form of a circle with a diameter of 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.
Answer:
The total wire required will be for 5 diameters and the circumference of the brooch.
The circumference of the brooch:-
=
=
Hence, the total wire required will be =
Q9: A brooch is made with silver wire in the form of a circle with a diameter of 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in Fig. 12.12. Find : (ii) the area of each sector of the brooch.
Answer:
The total number of lines present in the brooch is 10 (lines starting from the centre).
Thus, the angle of each sector is 36o.
The area of the sector is given by:
Answer:
It is given that the umbrella has 8 ribs, so the angle of each sector is 45o.
Thus, the area of the sector is given by:-
Hence, the area between two consecutive ribs is
Answer:
The area cleaned by one wiper is:-
Hence, the required area (area cleaned by both blades) is given by:-
=
=
Answer:
The area of the sector is given by:-
In this case, the angle is 80o.
Thus, the area is:-
Answer:
The angle of each of the six sectors is 60° at the centre. (∵
Area of the sector is given by:-
And the area of the equilateral triangle associated with the segment:-
=
Hence, the area of segment is: =
Thus, the total area of design is: =
So, the total cost for the design is: =
Q14: Tick the correct answer in the following :
Area of a sector of angle p (in degrees) of a circle with radius R is
(A)
(B)
(C)
(D)
Answer:
We know that the area of the sector is given by:-
=
Hence, option (d) is correct.
The topics discussed in the NCERT Solutions for class 10, chapter 11, Areas Related to Circles are:
The links below allow students to access all the Maths solutions from the NCERT book.
Also, read,
After completing the NCERT textbooks, students should practice exemplar exercises for a better understanding of the chapters and clarity. The following links will help students find exemplar exercises.
Students can check the following links for more in-depth learning.
Before planning a study schedule, always analyse the latest syllabus. Here are the links to the latest NCERT syllabus and some of the important books that will help students in this cause.
The length of an arc in a circle with radius r and central angle θ (in degrees) is:
Arc Length = $\frac{θ}{360°}×2πr
The area of a segment is found by subtracting the area of the triangle from the sector's area:
Segment Area = Sector Area − Triangle Area
Using the sector formula:
Sector Area = $\frac{θ}{360°}×πr^2$
The triangle's area can be determined using trigonometry or Heron's formula.
A sector is a portion of a circle (like a pizza slice). Its area is given by:
Area of sector = $\frac{θ}{360°}×πr^2$
where θ is the central angle in degrees.
Sector | Segment |
A sector includes a central angle and two radii. | A segment includes a chord and its corresponding arc. |
A sector looks like a pizza slice or wedge. | It is crescent-shaped. |
Centre of a circle is included in a sector. | Centre of a circle is not necessarily included in a segment. |
$\frac{θ}{360°}×πr^2$, where θ= central angle and r= radius | Segment Area |
Some of the common types of questions asked in this chapter are:
Hello
Since you are a domicile of Karnataka and have studied under the Karnataka State Board for 11th and 12th , you are eligible for Karnataka State Quota for admission to various colleges in the state.
1. KCET (Karnataka Common Entrance Test): You must appear for the KCET exam, which is required for admission to undergraduate professional courses like engineering, medical, and other streams. Your exam score and rank will determine your eligibility for counseling.
2. Minority Income under 5 Lakh : If you are from a minority community and your family's income is below 5 lakh, you may be eligible for fee concessions or other benefits depending on the specific institution. Some colleges offer reservations or other advantages for students in this category.
3. Counseling and Seat Allocation:
After the KCET exam, you will need to participate in online counseling.
You need to select your preferred colleges and courses.
Seat allocation will be based on your rank , the availability of seats in your chosen colleges and your preferences.
4. Required Documents :
Domicile Certificate (proof that you are a resident of Karnataka).
Income Certificate (for minority category benefits).
Marksheets (11th and 12th from the Karnataka State Board).
KCET Admit Card and Scorecard.
This process will allow you to secure a seat based on your KCET performance and your category .
check link for more details
https://medicine.careers360.com/neet-college-predictor
Hope this helps you .
Hello Aspirant, Hope your doing great, your question was incomplete and regarding what exam your asking.
Yes, scoring above 80% in ICSE Class 10 exams typically meets the requirements to get into the Commerce stream in Class 11th under the CBSE board . Admission criteria can vary between schools, so it is advisable to check the specific requirements of the intended CBSE school. Generally, a good academic record with a score above 80% in ICSE 10th result is considered strong for such transitions.
hello Zaid,
Yes, you can apply for 12th grade as a private candidate .You will need to follow the registration process and fulfill the eligibility criteria set by CBSE for private candidates.If you haven't given the 11th grade exam ,you would be able to appear for the 12th exam directly without having passed 11th grade. you will need to give certain tests in the school you are getting addmission to prove your eligibilty.
best of luck!
According to cbse norms candidates who have completed class 10th, class 11th, have a gap year or have failed class 12th can appear for admission in 12th class.for admission in cbse board you need to clear your 11th class first and you must have studied from CBSE board or any other recognized and equivalent board/school.
You are not eligible for cbse board but you can still do 12th from nios which allow candidates to take admission in 12th class as a private student without completing 11th.
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