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CBSE-i

Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India

(Core)

Areas Related

to Circles

Areas Related

to Circles

MATHEMATICS

UNIT-13

CLASS

X

MATHEMATICS

CBSE-i

UNIT-13

CLASS

X


(Core)

Areas Related

to Circles

MATHEMATICS

The CBSE-International is grateful for permission to reproduce

and/or translate copyright material used in this publication. The

acknowledgements have been included wherever appropriate and

sources from where the material may be taken are duly mentioned. In

case any thing has been missed out, the Board will be pleased to rectify

the error at the earliest possible opportunity.

All Rights of these documents are reserved. No part of this publication

may be reproduced, printed or transmitted in any form without the

prior permission of the CBSE-i. This material is meant for the use of

schools who are a part of the CBSE-International only.

The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step in making the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a fresh thought process in imparting a curriculum which would restore the independence of the learner to pursue the learning process in harmony with the existing personal, social and cultural ethos.

The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It has about 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board has always been conscious of the varying needs of the learners in countries abroad and has been working towards contextualizing certain elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged. The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in view.

The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand, appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations and additions wherever and whenever necessary.

The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to rethink their approaches for knowledge processing by their learners. In this context, it has become imperative for them to incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current, to upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of these requirements.

The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all subject areas to cater to the different pace of learners.

The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is now introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of CBSE-i is to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some non-evaluative components in the curriculum which would be commented upon by the teachers and the school. The objective of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives, SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this 'Core'. The Core skills are the most significant aspects of a learner's holistic growth and learning curve.

The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework (NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to millions of learners, many of whom are now global citizens.

The Board does not interpret this development as an alternative to other curricula existing at the international level, but as an exercise in providing the much needed Indian leadership for global education at the school level. The International Curriculum would evolve on its own, building on learning experiences inside the classroom over a period of time. The Board while addressing the issues of empowerment with the help of the schools' administering this system strongly recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to their peers through the interactive platforms provided by the Board.

I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and Dr. Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the development and implementation of this material.

The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion forums provided on the portal. Any further suggestions are welcome.

Vineet Joshi

Chairman

PREFACEPREFACE

ACKNOWLEDGEMENTSACKNOWLEDGEMENTSAdvisory Conceptual Framework

Ideators

Shri Vineet Joshi, Chairman, CBSE Shri G. Balasubramanian, Former Director (Acad), CBSE

Sh. N. Nagaraju, Director(Academic), CBSE Ms. Abha Adams, Consultant, Step-by-Step School, Noida

Dr. Sadhana Parashar, Director (Training),CBSE

Ms. Aditi Misra Ms. Anuradha Sen Ms. Jaishree Srivastava Dr. Rajesh Hassija

Ms. Amita Mishra Ms. Archana Sagar Dr. Kamla Menon Ms. Rupa Chakravarty

Ms. Anita Sharma Ms. Geeta Varshney Dr. Meena Dhami Ms. Sarita Manuja

Ms. Anita Makkar Ms. Guneet Ohri Ms. Neelima Sharma Ms. Himani Asija

Dr. Anju Srivastava Dr. Indu Khetrapal Dr. N. K. Sehgal Dr. Uma Chaudhry

Coordinators:

Dr. Sadhana Parashar, Ms. Sugandh Sharma, Dr. Srijata Das, Dr. Rashmi Sethi, Head (I and R) E O (Com) E O (Maths) E O (Science)

Shri R. P. Sharma, Consultant Ms. Ritu Narang, RO (Innovation) Ms. Sindhu Saxena, R O (Tech) Shri Al Hilal Ahmed, AEO

Ms. Seema Lakra, S O Ms. Preeti Hans, Proof Reader

Material Production Group: Classes I-V

Dr. Indu Khetarpal Ms. Rupa Chakravarty Ms. Anita Makkar Ms. Nandita Mathur

Ms. Vandana Kumar Ms. Anuradha Mathur Ms. Kalpana Mattoo Ms. Seema Chowdhary

Ms. Anju Chauhan Ms. Savinder Kaur Rooprai Ms. Monika Thakur Ms. Ruba Chakarvarty

Ms. Deepti Verma Ms. Seema Choudhary Mr. Bijo Thomas Ms. Mahua Bhattacharya

Ms. Ritu Batra Ms. Kalyani Voleti

English :

Geography:

Ms. Sarita Manuja

Ms. Renu Anand

Ms. Gayatri Khanna

Ms. P. Rajeshwary

Ms. Neha Sharma

Ms. Sarabjit Kaur

Ms. Ruchika Sachdev

Ms. Deepa Kapoor

Ms. Bharti Dave Ms. Bhagirathi

Ms. Archana Sagar

Ms. Manjari Rattan

Mathematics :

Political Science:

Dr. K.P. Chinda

Mr. J.C. Nijhawan

Ms. Rashmi Kathuria

Ms. Reemu Verma

Dr. Ram Avtar

Mr. Mahendra Shankar

Ms. Sharmila Bakshi

Ms. Archana Soni

Ms. Srilekha

Science :

Economics:

Ms. Charu Maini

Ms. S. Anjum

Ms. Meenambika Menon

Ms. Novita Chopra

Ms. Neeta Rastogi

Ms. Pooja Sareen

Ms. Mridula Pant

Mr. Pankaj Bhanwani

Ms. Ambica Gulati

History :

Ms. Jayshree Srivastava

Ms. M. Bose

Ms. A. Venkatachalam

Ms. Smita Bhattacharya

Material Production Groups: Classes IX-X

English :

Ms. Rachna Pandit

Ms. Neha Sharma

Ms. Sonia Jain

Ms. Dipinder Kaur

Ms. Sarita Ahuja

Science :

Dr. Meena Dhami

Mr. Saroj Kumar

Ms. Rashmi Ramsinghaney

Ms. Seema kapoor

Ms. Priyanka Sen

Dr. Kavita Khanna

Ms. Keya Gupta

Mathematics :

Political Science:

Ms. Seema Rawat

Ms. N. Vidya

Ms. Mamta Goyal

Ms. Chhavi Raheja

Ms. Kanu Chopra

Ms. Shilpi Anand

Geography:

History :

Ms. Suparna Sharma

Ms. Leela Grewal

Ms. Leeza Dutta

Ms. Kalpana Pant

Material Production Groups: Classes VI-VIII

1. Syllabus 1

2. Scope document 2

3. Teacher's Support Material 5

Teacher's Note 6

Activity Skill Matrix 10

Warm Up W1 11

Rangoli Design

Warm Up W2 12

Word Search

Pre Content P1 14

Drawing Pictures

Pre -Content P2 15

Introduction to Length of an Arc

Content Worksheet CW1 16

Length of an Arc


Area of a Sector


Area of a Segment


Simple Applications

Post Content Worksheet PCW1 20



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Content


Assessment Plan 21

4. Study Material 25

5. Student Support Material 39

SW1: Warm Up (W1) 40

Rangoli Design

SW2: Warm Up (W2) 42

Word Search

SW3: Pre Content (P1) 44

Drawing Pictures

SW4: Pre Content (P3) 46

Introducing Length of an Arc

SW5: Content (CW1) 48

Length of an Arc


Area of a Sector


Area of a Segment

SW8:Content (CW4) 64

Combination of Figures

SW9: Post Content (PCW1) 74

SW10: Post Content (PCW2) 75

SW 11: Post Content (PCW3) 77

SW 12: Post Content (PCW4) 79

Assignment

Suggested Videos & Extra Readings. 81

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1

SYLLABUS

AREAS RELATED TO CIRCLES

UNIT – 13 (CORE)

Introduction

Recall basic terms: perimeter, area

Perimeter and area related to circle

Circumference of circle, length of an arc, area of circle, area of segment, area of sector, area of semicircle

Application Simple Application problems- area enclosed between various circles/circular arcs.

2

SCOPE DOCUMENT

Key concepts/terms

1. Length of an arc

2. area of a sector

3. area of a segment

Learning objectives:

To explain all the terms related to circles.

To relate circle and its parts to real life.

To introduce the concept of length of an arc.

To verify the formula for the length of an arc using hands on activity and use it to

solve problems.

To find the formula for area of sector of circle and use it to solve problems.

To find area of a segment and to solve problems based on it.

To find the area enclosed between various circles or circular arcs.

EXTENSION ACTIVITY:

1. Draw a circle inscribed and circumscribed by a triangle. Find the ratio of area of

circle to inscribed triangle and ratio of area of circle to circumscribed triangle

2. Repeat the same activity for square.

3

Research:

What is the smallest radius for which the radius of the circular area covered by the five

equal disks placed symmetrically at the centre is 1.

CROSS-CURRICULAR LINK:

1. Circle graph depicts the relative amount of a whole. Each sector has area

proportional to the fraction of percentage represented by it.

2. Islamic art is based on designs which can be created by division of circles. Collect

and study some designs of Islamic art. Visit the following link to get some

interesting ideas on how to create designs using circles and other polygons

http://www.vam.ac.uk/content/articles/t/teachers-resource-maths-and-

islamic-art-and-design/

http://www.vam.ac.uk/content/articles/t/teachers-resource-maths-and-islamic-art-and-design/

http://www.vam.ac.uk/content/articles/t/teachers-resource-maths-and-islamic-art-and-design/

4

3.

Try to create this six circle symmetry. Create beautiful coloured patterns.

5

Teacher’s

Support Material

6

TEACHER’S NOTE The teaching of Mathematics should enhance the child’s resources to think and reason,

to visualize and handle abstractions, to formulate and solve problems. As per NCF

2005, the vision for school Mathematics includes:

1. Children learn to enjoy mathematics rather than fear it.

2. Children see mathematics as something to talk about, to communicate through, to

discuss among them, to work together on.

3. Children pose and solve meaningful problems.

4. Children use abstractions to perceive relationships, to see structures, to reason out

things, to argue the truth or falsity of statements.

5. Children understand the basic structure of Mathematics: Arithmetic, algebra,

geometry and trigonometry, the basic content areas of school Mathematics, all offer

a methodology for abstraction, structuration and generalisation.

6. Teachers engage every child in class with the conviction that everyone can learn

mathematics.

Students should be encouraged to solve problems through different methods like

abstraction, quantification, analogy, case analysis, reduction to simpler situations, even

guess-and-verify exercises during different stages of school. This will enrich the

students and help them to understand that a problem can be approached by a variety of

methods for solving it. School mathematics should also play an important role in

developing the useful skill of estimation of quantities and approximating solutions.

Development of visualisation and representations skills should be integral to

Mathematics teaching. There is also a need to make connections between Mathematics

and other subjects of study. When children learn to draw a graph, they should be

encouraged to perceive the importance of graph in the teaching of Science, Social

7

Science and other areas of study. Mathematics should help in developing the reasoning

skills of students. Proof is a process which encourages systematic way of

argumentation. The aim should be to develop arguments, to evaluate arguments, to

make conjunctures and understand that there are various methods of reasoning.

Students should be made to understand that mathematical communication is precise,

employs unambiguous use of language and rigour in formulation. Children should be

encouraged to appreciate its significance.

At the secondary stage students begin to perceive the structure of Mathematics as a

discipline. By this stage they should become familiar with the characteristics of

Mathematical communications, various terms and concepts, the use of symbols,

precision of language and systematic arguments in proving the proposition. At this

stage a student should be able to integrate the many concepts and skills that he/she has

learnt in solving problems.

This unit is extension of knowledge earned about the area of circle and other plane

figures, in previous classes. The unit contains hands on activities to reinforce previously

learnt concepts and lots of application problems with focus on following learning

objectives: Learning objectives:

To explain all the terms related to circles.



To verify the formula for the length of an arc using hands on activity and use it to

solve problems.

To find the formula for area of sector of circle and use it to solve problems.


To find area of a combination of figures and to solve problems based on it.

8

In this unit a lot of terms related to circle are used. Through warm up activity W1

students are given an opportunity to recognize all parts of circle and through activity

W2 they can recall all terms related to circle. Pre- content activity P1 tends to observe

the real-life object. These activities also help the learners to perceive the relation

between various parts of circle and the beautiful patterns emerging from them. This

type of observation skill will help them when they will attempt the problems involving

one or more circles or circles and other plane figures.

Pre-content activity P2 is preparation for making students understand the relation

between length of arc of circle and the circumference of the circle. First they will divide

the circles of given radius into three arcs of any length. For this purpose they can draw

circles on paper or they can have circular disks .they shall be asked to measure the

length of each arc and find the ratio of circumference to length of arc. But here they are

not supposed to use the formula of circumference/length of arc .They can measure the

respective lengths through thread and can tabulate their observations and conclusion.

Further they can validate or refine their conclusion by using the formula. Although the

students are familiar with the formula for circumference and length of arc, these

activities will strengthen the concept and will make the problem solving easy for them.

Similarily to establish the relation between area of sector and area of circle an

interesting activity is described in CW2. A pattern will emerge when the regular

polygons with different number of sides are cut into equal sectors. This activity in the

class shall be followed by rigorous discussions and the students shall be able to come

up with the formula themselves.

Same type of activity is described for area of segment. Activities are immediately

followed by the problem sheets as TASK 2.

9

Once again this is to remind all teachers that the purpose of the activities is not the

discovering of any new formula, but to strengthen the previous knowledge. Such

activities give insight to the students in problem solving of related topics.

When students start attempting the combination problems, the following three step

strategy can be adopted:

1. Identification/marking of required area as per requirement of problem

2. Word expression for area ,e.g. area marked yellow or shaded area in above figure is

ar(semicircle with radius 9/2 ) + ar (semicircle with radius 13.5)-ar(semicircle with

radius 4.5/2)

3. Algebraic expression for above word expression using formula

4. Calculations

10

ACTIVITY SKILL MATRIX

AREAS RELATED TO CIRCLES

Type of Activity

Name of Activity Skill to be developed

Warm UP(W1)

Rangoli Design Observation, Recall, Relate

Warm UP(W2)

Word Search Vocabulary Testing

Pre-Content (P1)

Drawing Pictures Relate to real context

Pre-Content (P2)

Introduction to length of an Arc

Observation, Analyze, Inference

Content (CW 1) Length of an Arc

Concrete to abstract, Application

Content (CW 2) Area of Sector Observation, Analyze, Inference, Application

Content (CW 3) Area of Segment

Thinking Skill, Problem Solving

Content (CW 4) Combination of Figures

Knowledge, Thinking Skills, Application

Post - Content (PCW 1)

Fill Ups Knowledge, Understanding


Open Ended Problem

Creative Skills, thinking skills


MCQ Conceptual knowledge, Problem Solving skills


Assignment Knowledge and application.

11

ACTIVITY 1- WARM UP W1

Rangoli Design

Specific Objective:

To revisit basic concepts of circles and its parts.

Description: This task is designed to revisit the concepts already learnt in earlier classes

related to circle. It will also be a fun exercise for students which give them the scope of

showing their creative skills while learning mathematical concepts.

Execution: Printed worksheets of rangoli designs may be distributed to the students.

Each student will colour the designs individually and will write the terms related to

circles.

Parameters for Assessment:

Able to recall basic terms related to circle.

Able to recognize various parts of circle

Able to appreciate the use of geometry in various designs

12

ACTIVITY 2- WARM UP W2

Word Search

Specific Objective:

To revise the terms related to circles.

Description: This is a word search task in which students will search the words related

to circle in the given worksheet and will encircle them. These words may be arranged

horizontally, vertically, diagonally in either way- left to right or right to left, top to

bottom or bottom to top. This task will help the students to recall all the terms related to

circle.

Execution: Printed worksheets may be distributed to the students. Sufficient time may

be given to the students to solve the worksheet. After the completion of task, students

may be shown the answer grid on projector so that they can check their answers.

13

Answer Grid:


Able to recall basic terms related to circle.

14

ACTIVITY 3- PRE CONTENT (P1)

Drawing Pictures

Specific Objective:


Description: In this task, students will make pictures of things around us which

corresponds to the words given in the sheet. For example, they can make a piece of

pizza corresponding to sector. This task is designed to make students appreciate the use

of geometry in our life.

Execution: Teacher may write the words on the board and students may draw the

corresponding pictures in their notebooks. Alternatively, printed worksheets may be

given to the students. A class discussion may be held to check the work of the students.


Able to recall and recognize circle & related terms.

Able to appreciate the use of mathematics all around us.

15

ACTIVITY 4 - PRE CONTENT (P2)


Specific Objective:


Description: In this task, students will draw circles with given radius. They will use

thread to measure the circumference of each circle. Further, students will divide the

circle in equal parts and measure the length of arc thus formed using thread. They will

also try to find out the relationship between circumferences with the length of an arc.

Execution: Teacher will explain the activity to the students and each student will

perform the activity in his notebook. Students will reflect on the statement given in the

worksheet in their notebooks based on their observation. A general class discussion

may be held thereafter.


Able to recall and recognize circumference and arc of a circle.

16

ACTIVITY 5- CONTENT (CW1)

Length of an Arc

Specific Objective:

To find the formula for the length of an arc and use it to solve problems.

Description: Task 1 is hands on activity to verify the formula for length of an arc. In this

task, students will find the arc length by actual measurement using thread and they will

also find out the length of the arc using formula. They will reflect their observations by

filling the observation table while doing the activity and will write the result of the

activity. Task 2 is based on the application of task 1. Students will solve questions based

on finding arc length when radius and central angle is given.

Execution: Each student will perform the activity individually and will record the

observations in their notebook. Printed worksheets may also be given in which students

will record their observation and write the result while performing the activity. Few

questions of task 2 may be taken as a black board task and the remaining may be solved

by the students in their notebooks.


Able to recall formula of circumference of a circle

Able to find circumference of a circle

Able to recall formula for arc length

Able to find length of an arc

Extra Reading:

http://www.onlinemathlearning.com/arc-circle.html


17


Area of a Sector

Specific Objective:

To find the formula for area of a sector of a circle and use it to solve problems.

Description: Task 1 is hands on activity to find the sum of areas of 3 sectors of same

radii. Students will follow the steps given in the task and will perform the activity. They

will extend the activity to polygons of 4 sides, 5 sides, 6 sides and so on. They will thus

try to find out whether a particular pattern is being generated. Task 2 is problem

solving task. The problems are based on finding area of sectors when some conditions

are given or vice versa.

Execution: Printed worksheets may be given in which students will record their

observation and write the result while performing the activity. Task 2 may be taken as a

black board task in which students may solve the questions on black board. All the

students will solve the questions in their notebook also.


Able to recall formula of area of a circle

Able to find area of a circle

Able to recall formula of a sector

Able to find area of a sector

Extra Reading:

http://attanolearn.com/excel/cbse-10th-math-areas-related-circles.jsf

http://link.brightcove.com/services/player/bcpid1842754532?bckey=AQ~~,AAAAA

GLvQak~,FI1DNn-0IPr4Wx6YGPHGjZVRrIIFDAUD&bctid=31871597001

http://www.onlinemathlearning.com/area-sector.html


http://link.brightcove.com/services/player/bcpid1842754532?bckey=AQ~~,AAAAAGLvQak~,FI1DNn-0IPr4Wx6YGPHGjZVRrIIFDAUD&bctid=31871597001



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Area of a Segment

Specific Objective:


Description: Task 1 is designed to generate the formula for finding area of segment

using area of triangle and area of sector. While solving the step by step questions given

in the task, students will find the formula for area of a segment. Task 2 contains

questions based on finding area of a segment.

Execution: Printed worksheets may be distributed for both the tasks. But after

completion of task 1, a class discussion may be held. Teacher should explain a few

questions of task 2 on board before distributing the worksheets for task 2.


Able to recall area of a segment

Able to find area of a segment

Extra Reading:

http://link.brightcove.com/services/player/bcpid1842754532?bckey=AQ~~,AAAAA

GLvQak~,FI1DNn-0IPr4Wx6YGPHGjZVRrIIFDAUD&bctid=37227603001



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Simple Applications

Specific Objective:


Description: Task 1 gives students opportunity of using their creative skills to make

figures. This will also make them feel that geometry is used in their day to day life.

Students will also find the area of regions with different colours (used by them in

figures made by them). Task 2 is problem solving task which involves the use of area of

sectors and segments.

Execution: Task 1 is to be done by the students in their notebooks. Teacher may

describe the task and may give it as a home task by showing them the example given in

the worksheet. Few problems given in task 2 should be done by the students on black

board one by one followed by discussions. Remaining problems may be solved by the

students individually.


Able to find area of combination of figures

Able to solve real life problems of area related to circles.

Extra Reading:

http://www.authorstream.com/Presentation/WiZiQ-76444-Areas-related-circles-chap-

12-Education-ppt-powerpoint/

http://www.elcues.com/help/classX/XM12/

http://www.authorstream.com/Presentation/WiZiQ-76444-Areas-related-circles-chap-12-Education-ppt-powerpoint/



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ACTIVITY 9- POST CONTENT (PCW1)

Students will be assessed on the worksheet containing fill ups on basic concepts.


Assessment of the students will be done on question framing skills.

ACTIVITY 11-POST CONTENT (PCW3)

Assessment of the students will be done by using MCQs.


Students will be assessed on the worksheet containing questions.

21

ASSESSMENT PLAN

Assessment guidance plan for teachers

With each task in student support material a self–assessment rubric is attached for students. Discuss with the students how each rubric can help them to keep in tune their own progress. These rubrics are meant to develop the learner as the self motivated learner.

To assess the students’ progress by teacher two types of rubrics are suggested below, one is for formative assessment and one is for summative assessment.

Suggestive Rubric for Formative Assessment (exemplary)

Parameter Mastered Developing Needs motivation

Needs personal attention

Area of combina-tion figures

This student is able to

Identify/mark the required area correctly


Express in words the required area correctly


write the formula for each part correctly


Substitute each value correctly

Calculation are accurately done


Identify/mark the required area Correctly





Not able to

substitute each value correctly

Calculations are not correctly do me


Identify/ mark the required area correctly



Not able to


Calculations are not correctly do me

Not able to Identify/mark the required area correctly

22

From above rubric it is very clear that

Learner requiring personal attention is poor in concepts and requires the training of

basic concepts before moving further.

Learner requiring motivation has basic concepts but face problem in calculations or in

making decision about suitable substitution etc. He can be provided with remedial

worksheets containing solution methods of given problems in the form of fill-ups.

Learner who is developing is able to choose suitable method of solving the problem and

is able to get the required answer too.

Learner who has mastered has acquired all types of skills can be given more

challenging problems..

TEACHERS’ RUBRIC FOR SUMMATIVE ASSESSMENT OF

THE UNIT

Parameter 5 4 3 2 1

Recalling basic terms

Able to explain all the terms like circumference, area of circle, arc, arc length, area of segment, area of sector ,area of semicircle

Able to tell formulae of all above terms correctly

Able to use the formulae and can find the required area or circumference or length correctly

Not able to explain any term like circumference , area of circle, arc, arc length, area of segment, area of sector , area of semicircle

Not able to tell formulae of all the terms stated above correctly

Not able to use the formulae and cannot find the required

23

area or circumference or length correctly

Area related to circle

Able to find the required area of circle, area of sector, area of segment or circumference or length of arc , correctly

Able to find radius and angle of sector using the given information correctly.

Able to solve real life problems

Not able to find the required area of circle, area of sector, area of segment or circumference or length of arc , correctly

Not able to find radius and angle of sector using the given information correctly.

Not able to solve real life problems

Area of combination of figures

Able to identify the area required when a figure is combination of circle and other plane figures.

Able to find area mentioned above using appropriate formulae correctly

Not able to identify the area required when a figure is combination of circle and other plane figures.

Not able to find area mentioned above using appropriate formulae correctly

24

Study

Material

25

AREAS RELATED TO CIRCLES (Core)

INTRODUCTION

You are already familiar with the concept of a circle and some basic terms such as

centre, radius, arc, chord etc related to a circle. You have also learnt to find the

perimeter and area of a plane figure like square, rectangle, quadrilateral such as a

trapezium, parallelogram, rhombus, triangle etc. You also know how to find area and

circumferance (perimeter) of a circle. In this unit, we shall first briefly recall these

concepts related to a circle and extend the study to find areas of a sector and a segment

of a circle.

We shall also find areas of combinations of figures involving a circle and its parts.

1. Circumference and Area of a circle: A recall

Circumference

Recall that the distance covered by going around a circle one time is called its

perimeter or circumference.

You also know that

circumference of a circle

diameter

is a constant, denoted by a Greek letter 𝜋 (read as ―pi‖).

or circumference of a circle

diameter= 𝜋

or circumference = 𝜋 × diameter

= 𝜋 × 2r,

where r is the radius of the circle.

26

𝜋 is an irrational number but for calculation purposes, its value is usually taken as 22

7 or

3.14. The great Indian mathematician. Aryabhatta (AD. 476-550) gave an approximate

value of 𝜋 as 𝜋 = 62832

20000 which is nearly equal to 3.1416.

Great Indian mathematician S.Ramanujan (1887-1920) also calculated the value of 𝜋 to

some decimal places using an identity.

Area of a Circle

Recall that area of a circle of radius r is 𝜋r2, i.e.,

Area of a circle = 𝜋r2

Let us consider some examples:

Example 1: The radii of two circles are 6cm and 8cm. Find the radius of the circle having

its area equal to the sum of the areas of the two circles.

Solution: Let r1 = 6cm, r2 = 8cm.

Area of the circle with radius 𝑟1 = 𝜋𝑟12 = 𝜋(6)2cm2 = 36 𝜋cm2

Area of the circle with radius 𝑟2 = 𝜋𝑟22 = 𝜋(8)2cm2 = 64 𝜋cm2

Area of new circle = 𝜋R2 = 36 𝜋 + 64 𝜋 = 100 𝜋cm2,

where R is the radius of the new circle.

Thus 𝜋R2 = 100 𝜋

or, R2 = 100

or, R = 10

Hence, the required radius = 10cm.

27

Example 2: The radii of two circles are 12cm and 21cm. Find the radius of the circle

which has circumference equal to the sum of the circumference of the two circles.

Solution: Let r1 = 12cm, r2 = 21cm.

Circumference of the circle with radius r1 = 2 𝜋 r1 = 2 𝜋 (12) = 24 𝜋cm

Circumference of the circle with radius r2 = 2 𝜋r2

= 2 𝜋(21)

= 42 𝜋 cm

Circumference of the new circle = 24 𝜋 + 42 𝜋

= 66 𝜋

(where R is the radias of the new circle)

Thus, 2 𝜋 R = 66 𝜋

or, R=33

i.e., required radius = 33cm

Example 3: Find the cost of ploughing circular field whose circumference is 176m at the

rate of Rs.5 per square metre.

Solution: Let the radius of the field be r metres.

Then, 2 𝜋 r = 176

or r = 176 × 7

2×22 = 28m

∴ Area of the field = 𝜋r2

= 22

7 x 28 x 28 m2

28

= 2464 m2

Cost of ploughing the field at the rate of Rs 5 per m2

= Rs 5 x 2464

= Rs 12320

2. Sector of a circle

Recall that the portion (or part) of the circular region enclosed by the two radii and

the arc between the two ends of radii is called a sector of the circle. (See Fig.1

FIG. 1

In the figure, O is the centre, OA and OB are two radii, APB is an arc between the two

ends A and B of the two radii.

The shaded part OAPB is a sector. Unshaded portion OAQB is also a sector.

Clearly, the sector AOBP is a Minor Sector and the sector OAQB is a

Major Sector. ∠AOB is called angle of the sector.

Thus θ is the angle of the minor sector and 360o- θ is the angle of the major sector.

Area of a sector

You know that area of a circle of radius r is 𝜋r2.

29

You can imagine the circular region formed by the circle of radius r as a sector of

angle 360o (Because angle at the centre is a complete angle).

With this assumption, we can calculate the area of the sector OAPB as follows:

Area of a sector of angle 360o = 𝜋r2

So, area of a sector of angle 1o = 𝜋𝑟2

360o

Hence, area of a sector of angle θ0 =

𝜋𝑟2

360o × θ0

= 𝜋𝑟2θ

360

Length of the Arc of a sector

You know that circumference of a circle of radius r is 2𝜋r.

You can calculate the length of the arc of sector OAPB as follows:

Length of the arc of a sector of angle 360o = 2𝜋r

So, length of the arc of a sector of angle 1o = 2πr

360o

Hence, length of the arc of a sector of angle θ0 =

2πr θ

360

We now take some examples to illustrate the applications of the above formule.

Example 4: Find the area of a sector of a circle with radius 14cm and of angle 45o. Also, find the length of the corresponding arc of the sector.

Solution: Area of the sector = πr2θ

360

=22

7

× 14 × 14 ×45

× 360 cm2

= 11 x 7 cm2 = 77cm2

Length of the arc = 2𝜋𝑟 θ

360

= 2 x 22

7

× 14 ×45

× 360 cm

30

= 11cm

Example 5: In a circle of diameter 42cm, an arc subtends an angle of 60o at the centre.

Find:

(i) Length of the arc.

(ii) Area of the corresponding sector.

(iii) Area of the corresponding major sector.

(iv) Length of the major sector.

Solution: (i) Length of the arc = 2πr θ

360

= 2 x 22

7

× 21 ×60

× 360 (Diameter = 42cm, so, r =

42

2 = 21)

= 22cm

(ii) Area of the sector = 𝜋𝑟2 θ

360

= 22

7

× 21 ×21 × 60

× 360

= 231 cm2

(iii) Area of the major sector = 𝜋𝑟2 (3600−θ)

3600

= 22

7

× 21 ×21 × 3600− 600

× 360o

= 22 x 3 x 21 x 3000

3600

= 11 x 21 x 5 cm2 = 1155 cm2

Alternate Method: Area of the major sector

= area of the circle – area of minor sector

= 22

7 × 21 × 21 − 231 cm2

31

= (1386 – 231) cm = 1155 cm2

(v) Length of the major sector = 2𝜋𝑟 (3600−θ)

3600

= 2 x 22

7 x 21 x

(3600−600)

3600

= 2 x 22 x 3 x 300

3600

= 22 x 5 = 110 cm

Alternate Method: Length of the major sector

= circumference of the circle length of minor sector

= (2 x 22

7 x 21 – 22) cm

= (132 – 22) cm = 110cm

Example 6: A car has two wipers which do not overlap. Each wiper has a blade of

length 28cm sweeping through an angle of 120o. Find the total area cleaned at each

sweep of the blades.

Solution: Area swept by one wiper = Area of sector of central angle 120o

= 𝜋𝑟2 θ

360

= 22

7 x 28 x 28 x

120

360 cm2

So, total area swept by two wipers

= 2 x 22

7 x 28 x 28 x

120

360 cm2

= 4928

3 = 1642.67 cm2

32

Fig. 2

3. Area of segment of a circle

Recall that a chord of a circle divides the circular region into two parts. Each part is called a segment of the circle.

FIG. 3

In the figure, APB is the minor segment and AQB is the major segment

To find area of the minor segment APB, join the centre O to A and B.

Let ∠AOB = θ.

Area of minor segment APB

= Area of sector OAPB

— Area of ∆OAB

= πr2 θ

360 - Area of ∆OAB

Similarly,

Area of major segment AQB

33

FIG. 3

= Area of sector OAQB + area of ∆ OAB

= πr2(3600 – θ)

3600 + area of ∆ OAB

Alternatively

Area of major segment AQB

= Area of circle with centre O Area of minor segment APB.

Let us illustrate the applications of these formulas through some examples.

Example 7: Find the area of the segment APB shown in Fig 4, if radius of circle is 14cm

and the central angle is 60o.

Also, find the area of the corresponding major segment.

FIG. 4

34

Solution: Area of sector OAPB

= πr2θ

360 =

22

7

× 14 × 14 ×60

× 360

= 308

3 cm2

Area of ∆OAB = 𝟑

𝟒 r2 [∆OAB is an equilateral triangle as OA = OB. Area of an

equilateral triangle = 3

4 a2]

= 3

4 x 14 x 14

= 49 3 cm2

So, the area of segment APB = 308

3 49 3

= 102.6 84.87

= 17.80 cm2

Note: We can also find area of ∆OAB as follows:

(i) Area of ∆OAB = 1

2. AB x altitude

FIG. 5

Now, AC = r sin 30o = r×1

2 =

𝑟

2]

Also, AB = 2AC = r

So, OC = r sin 60o = r x 3

2

35

Area of ∆OAB = 1

2 AB x altitude

= 1

2 (r) r×

3

2

= 1

2 x 14 x 14 x

3

2

= 49 3 cm2

Area of corresponding major segment AQB

= Area of major sector OAQB

+ area of ∆OAB

= πr2(3600 – θ)

3600 + 49 3 cm2

= 22

7 × 14 × 14 ×

3000

3600 + 49 3 cm2

= 22

7 × 14 × 14 ×

5

6 + 49 3 cm2

= 1540

3+ 49 3 cm2

= 513.33 + 84.87 = 598.20 cm2

36

Alternate Method

Area of major segment AQB = Area of the circle — area of minor segment

= 𝜋r2 308

3+ 49 3 cm2

= 22

7 x 14 x 14

308

3+ 49 3 cm2

= 616 − 308

3+ 49 3 cm2

= 1540

3+ 49 3 cm2

Example 8: A chord of a circle of radius 10cm subtends a right angle at the centre. Find the area of

(i) Minor segment

(ii) Major segment (use 𝜋 = 3.14)

Solution:

(i) Area of minor segment APB

FIG. 6

37

= area of sector OAPB - area of ∆AOB

= πr2θ

3600 –

1

2 OA x OB

= (3.14) x 10 x 10 x 900

3600

1

2x 10 x 10

= (78.50 – 50) cm2

= 28.50 cm2

(ii) Area of major segment AQB

= area of circle- area of minor segment

= (3.14 x 10 x 10 – 28.50) cm2

= (314 – 28. 50) cm2

= 288.5 cm2

4. Areas of Combination of Plane Figure and Circles.

In daily life, we see many designs which involve circles along with other plane

figures such as square, triangle, rectangle etc. We now illustrate the process of

calculating areas of such figures/ designs through some examples.

Example 9: Find the area of a flower bed with semicircular ends as shown in Fig.7:

FIG. 7

(Use 𝜋 = 3.14)

38

Solution: The flower bed consists of a rectangle of dimensions 38cm x 10cm and two

semicircles each of radius 10cm.

Fig. 8

So, area of the flower bed

= area of the rectangle + area of two semicircles

= [38 x 10 + 1

2 𝜋 (5)2 +

1

2 𝜋 (5)2] cm2

= [380 + 3.14 x 25] cm2

= (380 + 78.5) cm2

= 458.5 cm2

39

Student’s

Support Material

40

STUDENT’S WORKSHEET 1

WARM UP (W1)

Rangoli Design

Name of Student___________ Date________

Observe the following rangoli pattern

Write as many words as you can, related to circle, by observing the above pattern.

Trace out the above pattern and colour it to make a beautiful rangoli design.

41

SELF ASSESSMENT RUBRIC 1 – WARM UP (W1)

Parameter

Able to recall basic terms

related to circle.

Able to recognize various

parts of circle

Able to appreciate the use

of geometry in various

designs

42


WARM UP (W2)

Word Search


The following picture contains words having terms related to circle. These words may

be arranged horizontally, vertically, diagonally in either way- left to right or right to

left, top to bottom or bottom to top. Find them and encircle them.

43

SELF ASSESSMENT RUBRIC 2 – WARM UP (W2)

Parameter

Able to recall basic terms

related to circle.

44


PRE CONTENT (P1)

Drawing Pictures


Draw pictures of real time objects corresponding to each word:

Object Word

circle

semicircle

Arc

sector

Segment

45

SELF ASSESSMENT RUBRIC 3 – PRE CONTENT (P1)

Parameter

Able to recall and

recognize circle & related

terms

Able to appreciate the use

of mathematics all around

us.

46


PRE CONTENT (P2)



Draw five circles of radii 5cm, 10cm, 15cm, 20cm and 25 cm. Measure their

circumference using a thread. Divide each circle in 3 equal parts. Measure the length of

arc for each part using a thread.

Is there any relation between the circumference and the arc lengths? Reflect.

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47

SELF ASSESSMENT RUBRIC 4 – PRE CONTENT (P2)

Parameter

Able to recall and

recognize circumference

and arc of a circle

48


CONTENT WORKSHEET (CW1)

Length of an Arc


Task 1: Hands on Activity:

Aim: To verify the formula for length of an Arc.

Material Required: Paper, compass, pencil, thread, measuring scale, pair of scissors.

Previous Knowledge: Finding Circumference

Procedure:

1. Take a circle of radius 5 units.

2. Divide the circle in 10 equal parts.

3. Measure the arc length of each part using thread.

4. Calculate the length of arc using the formula 𝜋𝑟𝜃

180˚

5. Repeat the above steps taking different radii.

Observation:

Radius Arc length by measurement Arc length by calculation

Result:

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49

Task 2:

Do the following:

1. In a circle, find the length of a 30⁰ arc if the circumference is 60 𝜋.

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2. Find the arc intercepted by the side of a regular pentagon inscribed in a circle of

radius 3 cm.

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50

3. Find the measure of central angle of an arc whose length is 3m if the circumference

is 9 m.

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4. In a circle, find the length of a 90⁰ arc if an inscribed hexagon has a side of length 12

cm.

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51

5. Find the length of a 180⁰ arc if the circumference is 25 units.

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52

SELF ASSESSMENT RUBRIC 5


Parameter

Able to recall formula of circumference of a circle

Able to find circumference of a circle

Able to recall formula for arc length

Able to find length of an arc

53



Area of a Sector


Task 1: Hands on Activity:

Sum of areas of 3 sectors of same radii

Aim:

By paper cutting and pasting, verify that the sum of areas of three sectors of same

radius ―r‖ formed at the vertices (as center) of any triangle is (Pi x r ^2)/2. [𝜋.𝑟2

2]

Material Required

Coloured paper, pair of scissors, glue, and geometry box.

Procedure

Step 1 Draw an equilateral triangle ABC on a coloured paper and cut it.

Step 2 With suitable radius, draw three sectors on the three vertices.

Step 3 Cut the three sectors.

Step 4 Draw a straight line and place the three sector cut outs adjacent to each other on

it.

Step 5 Write the observations.

Step 6 Repeat the activity for two more triangles other than equilateral triangle.

Step 7 Write the observations

Observations

1. When sector cut outs are placed on a straight line, they completely cover the straight

angle. So their sum is 180 degrees.

2. The three sectors cut outs placed on a straight-line form a semicircle.

3. The area of semicircle is Pi x r^2 /2.

http://mykhmsmathclass.blogspot.com/2007/12/class-x-activity-15.html

54

Source:

http://mykhmsmathclass.blogspot.com/search/label/ActivitySum%20of%20areas%20

of%203%20sectors%20of%20same%20radii%20cut%20at%20vertices%20of%20a%20trian

gle

Extend the above activity to find the sum of areas of

4 sectors of same radii cut out from 4 side regular polygon




Do you observe any pattern?

http://mykhmsmathclass.blogspot.com/search/label/ActivitySum%20of%20areas%20of%203%20sectors%20of%20same%20radii%20cut%20at%20vertices%20of%20a%20triangle



55

Task 2: Do the following

1. Find the are of a 60⁰ sector if the circumference is 10 𝜋.

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2. Find the area of a 240⁰ sector if the area of the circle is 60 sq. units.

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56

3. Find the measure of a central angle of an arc whose length is 10 𝜋 if the area of the

sector is 50 𝜋 sq units.

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4. Find the radius of a circle if a sector of area 24 𝜋 sq units has an arc length of 12 𝜋

units.

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57

5. Find the radius of a circle if 270⁰ sector has an area of 9 𝜋 sq units.

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58



Parameter

Able to recall formula of area of a circle

Able to find area of a circle

Able to recall formula of a sector

Able to find area of a sector

59



Area of a Segment


Task1: Finding area of a segment

Observe the following figure carefully.

Find XP in terms of r and θ.

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Write PQ in terms of r and θ.

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60

Find OX in terms of r and θ.

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Find area of triangle POQ in terms of r and θ.

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Find area of sector POQ.

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Find area of the segment.

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Task 2:

Do the following:

1. Find the area of segment of a circle if the radius of the circle is 12 units and the

central angle measures 90⁰.

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61

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2. Find the area of segment of a circle if the radius is 6 cm and the central angle

measures 60⁰.

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3. Find the area of segment of a circle if the radius of the circle and the chord of the

segment each have length 9 units.

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62

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4. A chord of length 12 units is 5 units from the centre of the circle. Find the area of the

segment.

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5. An arc length of 4 𝜋 inscribes an angle of 90⁰ at the centre. Find the area of

corresponding segment.

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63

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Parameter

Able to recall formula for area of a segment

Able to find area of a segment

64



Combination of Figure

Simple Applications


Task 1:

Sajeev made the following shape using circles and semicircles only.

Draw a joker using circles, semicircles, arcs, sector and segments only. Colour your

joker with different colours.

65

Find the area under different colour regions.

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Task 2:

Find the area under different colour regions in the following figures:

1.

66

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2.

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67

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3.

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68

4.

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69

5.

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70

6.

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7.

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8.

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73



Parameter

Able to find area of combination of figures

Able to find area related to circles.

74


POST CONTENT (PCW1)


Fill in the blanks.

(a) The perimeter of a circle is called its _________________.

(b) 𝜋 is the ratio of _______________ of a circle with its ____________.

(c) 𝜋 is an _______________ number.

(d) The approximate value of 𝜋 is ____________ .

(e) Area of the sector of a circle of angle θ = ______________.

(f) Length of an arc of a sector of a circle of angle θ = __________.

(g) The length of an arc of a circle of radius, subtending angle θ at the centre is

_________.

(h) The area of the sector of a circle of radius r with central angle θ is =___________.

(i) The perimeter of sector of a circle of radius r and central angle θ is _____________.

75


POST CONTENT (PCW2)


Write word problems based on the figures given below. Your questions should be

related to day to day life:

77


POST CONTENT (PCW3)


Multiple Choice Questions

1. The area of the sector of a circle of radius r and central angle θ, is

A. ½ l.r B. 2𝜋r2θ/720 C. 2𝜋rθ/360 D. 𝜋rθ/360

2. An arc of a circle is of length 5 𝜋 cm and the sector it bounds has an area of 20 𝜋 cm2.

The radius of circle is

A. 1 cm B. 5 cm C. 8 cm D. 10 cm

3. A sector is cut from a circle of radius 21 cm. The angle of sector is 150º. The area of

sector is

A. 577.5 cm2 B. 288.2 cm2 C. 152 cm2 D. 155 m2

4. A chord AB of a circle of radius 10 cm makes a right angle at the centre of the circle.

The area of major segment is

A. 210 cm2 B. 285.7 cm2 C. 185.5 cm2 D. 258.1 cm2

5. A horse is tied to a pole with 56 m long string. The area of the field where the horse

can graze is

A. 2560 m2 B. 2464 m2 C. 9856 m2 D. 25600 m2

6. The circumferences of two circles are in the ratio 2:3. The ratio of their areas is

A. 4:9 B. 2:3 C. 7:9 D. 4:10

7. Area enclosed between two concentric circles is 770 cm2. If the radius of outer circle is

21 cm, then the radius of inner circle is

A. 12 cm B. 13 cm C. 14 cm D. 15 cm

78

8. The perimeter of a semi-circular protector is 72 cm. Its diameter is

A. 28 cm B. 14 cm C. 36 cm D. 24 cm

9. The minute hand of a clock is 21 cm long. The area described by it on the face of clock

in 5 minutes is

A. 5.5 cm2 B. 2.5 cm2 C. 1.5 cm2 D. 3.5 cm2

10. The area of a circle circumscribing a square of area 64 cm2 is

A. 50.28 cm2 B. 25.5 cm2 C. 100.57 cm2 D. 75.48 cm2

79


POST CONTENT (PCW4)

ASSIGNMENT


Solve the following problems:

1. Find the length of an arc of radius 14 cm and central angle

a) 30⁰.

b) 60⁰.

c) 120⁰.

d) 150⁰.

2. Find the area of a sector with radius 7 cm and central angle

a) 30⁰.

b) 60⁰.

c) 120⁰.

d) 150⁰.

3. Find the area of a segment with radius 21 cm and central angle

a) 30⁰.

b) 60⁰.

c) 120⁰.

d) 150⁰.

4. Find the central angle corresponding to an arc whose radius is 3.5 cm and arc length

is 16 cm.

5. The length of an arc subtending an angle of 720 at the center is 44 cm. Find the area

of the circle.

80

6. Find the area of a sector of radius 7 cm and central angle 30⁰.

7. Find the area of a segment of a circle of radius 14 cm and arc length 7𝜋.

8. Find the area under colour region in the given figure.

81

SUGGESTED VIDEOS AND EXTRA READINGS:



http://link.brightcove.com/services/player/bcpid1842754532?bckey=AQ~~,AAA

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http://link.brightcove.com/services/player/bcpid1842754532?bckey=AQ~~,AAA

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http://www.authorstream.com/Presentation/WiZiQ-76444-Areas-related-circles-

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