Circles

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Class 9 Mathematics contain simple description and theorems on circles. 9 Mathematics contain higher level description and theorems on circles.

Additional Information

Useful websites

  1. maths is fun A GOOD WEBSITE ON DEFINITIONS FOR CIRCLES
  2. COOL MATH GIVES CLEAR AND EASY DEFINITIONS
  3. OPEN REFERENCE CONTAINS FEW SIMULATIONS
  4. WIKIPEDIA CONTAINS EXPLANATIONS FOR CIRCLES
  5. KHAN ACADEMY CONTAIN GOOD VIDEOS
  6. ARVIND GUPTA TOYS CONTAIN VERY GOOD BOOKS ON MATHEMATICS AND SCIENCE
  7. nrich.maths.org Contain very good description of Pie

Reference Books

  1. School Geometry By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.

Teaching Outlines

Introduction

Concept #1 CIRCLE

Notes for teachers

Introduction: The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.

Learning objectives

  1. Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.
  2. To make students know that circle is a 2-dimensional plane circular figure.
  3. All points on its edge are equidistant from the center.
  4. The method of drawing a circle
  5. The size of the circle is defined by its radius.
  6. To elicit the difference between a bangle or a circular ring and circle as such.


 


Activity No # 1. A discussion on “Life without wheels / circular shaped figures.”

  • Estimated Time: 45 minutes
  • Materials/ Resources needed: Paper, pen
  • Prerequisites/Instructions, if any:

Previous day homework :

  1. Ask the children to make a list of all circular objects that they can think of :
  2. List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)
  3. Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.
  • Multimedia resources
  • Website interactives/ links/ / Geogebra Applets
  • Process:(How to do the activity)

Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.

  • Developmental Questions :(What discussion questions)
  1. What all shapes do we see around us ?
  2. Can you imagine bicycles and your other vehicles without circular wheels ?
  3. How different life would have been if wheel was not disovered ?
  4. How could we explain centrifugal force ?
  5. What about potter's wheel and stone mill?
  6. Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?
  • Evaluation:
  1. Do you all now agree that wheel is one of the greatest inventions of mankind? Justify
  • Question Corner:
  1. Are shapes important ? How?
  2. Is bangle a circle ?
  3. Would it be possible to understand pi without understanding circles ?


Activity No #2. Circle of varying radius using Geogebra

  • Estimated Time: 20 mins
  • Materials/ Resources needed: Laptop, geogebra,projector and a pointer
  • Prerequisites/Instructions, if any:
  1. The students should know a circle
  2. They should know the meaning of radius .
  3. They should know to measure radius using compass
  4. They should know to draw a circle of given radius using compass.
  • Multimedia resources: Laptop
  • Website interactives/ links/ / Geogebra Applets:

  • Process:
  1. Review the terms plane closed figures, circle and radius.
  2. Show them how to measure radius on the compass
  3. Show them to draw a circle.
  4. Project the geogebra and let them view the changing size of the circle with changing radius.
  • Developmental Questions:
  1. What is a circle ?
  2. Which point is the centre of the circle ?
  3. What is the radius of this circle ?
  4. How do you name the radius ?
  • Evaluation:
  1. By what parameter is the size of a circle defined ?
  2. Bigger the radius, _____________ is the size of the circle.
  • Question Corner:
  1. How do you name a circle ?
  2. Can you draw a circle without knowing the radius ?

Activity No # 3.Is circle a Polygon ? - A debate.

  • Estimated Time: 40 minutes

*Materials/ Resources needed Laptop, geogebra file, projector and a pointer.

  • Prerequisites/Instructions, if any
  1. The students should know that the plane figures with 3 or more than 3 sides are called polygons.
  2. They should know that as the number of sides increases, the figure and hence its name also changes.
  3. They should know the basics of a circle.

*Multimedia resources Laptop

  • Website interactives/ links/ / Geogebra Applets

  • Process
  1. Ask the students what polygons are ?
  2. As you increase the number of sides ask them to count the number of sides.
  3. Ask them the name of the shape formed.
  4. As the number of sides increase,let them speak about the length of the sides ?

*Developmental Questions

  1. How many sides does this figure have ?
  2. Name the figure formed.
  3. What is hapenning to the length of the sides as the number of sides is increased ?
  4. What shape is this ?
  5. So, can circle be considered a polygon ? Justify
  • Evaluation:
  1. Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?
  2. Are the students able to appreciate the application of polygon anology to circles.
  • Question Corner

Debate between two groups with these two perspectives.

  1. Circle seems to have derived from polygons . Circle can be considered a polygon.

Vs

  1. A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)

Activity No #

  • Estimated Time
  • Materials/ Resources needed
  • Prerequisites/Instructions, if any
  • Multimedia resources
  • Website interactives/ links/ / Geogebra Applets


  • Process/ Developmental Questions
  • Evaluation
  • Question Corner

Concept # 2. Measurements in circles

Learning objectives

  1. The students should learn to measure radius, diameter, circumference, chord length and angles subtended at the centre and on the circumference of the circle.
  2. The students should understand that radius, diameter and chord lengths are linear measurements.
  3. They should learn to relate the size of the circle with radius.
  4. They realise that to draw a circle knowing the measure of radius or diameter is essential.
  5. There can be infinite radii in a circle.
  6. Diameter is twice the radius.
  7. The students should understand what a chord is.
  8. Chords of different lengths can be drawn in a circle.
  9. Chord length can be measured using a scale and its units is cm.
  10. They should know that the length of the chord increases as it moves closer to the diameter.
  11. The longest chord in the circle is its diameter.
  12. Distance of chord from the centre is its perpendicular distance from the centre.
  13. A chord divides the circle into two segments.
  14. Angle at the centre of the circle is 360º.
  15. Angles in circles are measured using protractor.
  16. Circumference and area are calculated using formula.

Notes for teachers

Activity No # 1. Measuring radius and diameter.

  • Estimated Time: 15 mins
  • Materials/ Resources needed:
  1. Laptop, goegebra tool, projector and a pointer.
  2. students' geometry box
  • Prerequisites/Instructions, if any:
  1. Children should have the knowledge of circle, centre, radius, diameter and circumference.
  2. The teacher should have the necessary skill of using geogebra tool.
  • Multimedia resources: Laptop
  • Website interactives/ links/ / Geogebra Applets

  • Process:
  1. Initially the teacher can explain the terms: circle, its centre, radius, diameter and circumference.
  2. Ask the children “What parameter is needed to draw a circle of required size ?”
  3. Show them how to measure radius on the scale accurately using compass.
  4. Show them to draw a circle.
  5. Given diameter, radius = D/2.
  6. Also the other way i.e. If a circle is given, then its radius can be measured by using scale which is the linear distance between centre of the circle and any point on the circumference.
  7. To measur diameter, measure the length of that chord which passes through the centre of the circle.

Then she can project the digital tool 'geogebra.' and further clarify concepts.

  • Developmental Questions:
  1. Name the centre of the circle.
  2. Name the point on the circumference of the circle.
  3. What is the linesegment AB called ?
  4. Name the line passing through the centre of the circle.
  5. Using what can you measure the radius and diameter.
  6. Name the units of radius/diameter.
  • Evaluation:
  1. How do you measure exact radius on the compass?
  2. Are the children able to corelate the radius/diameter of a circle with its size ?
  • Question Corner:
  1. If the centre of the circle is not marked , then how do you get the radius for a given circle.
  2. How many radii/diameter can be drawn in a circle?
  3. Are all radii for a given circle equal ?
  4. Is a circle unique for a given radius/diameter ?
  5. In how many parts does a diameter divide the circle ? What is each part called ?

Activity No # 2 Measuring a chord in a circle.

  • Estimated Time : 10 minutes
  • Materials/ Resources needed:

Laptop, geogebra file, projector and a pointer.

  • Prerequisites/Instructions, if any:
  1. The students should have prior knowledge of circle, radius , diameter and circumference..
  2. The teacher should have knowledge of using geogebra.
  • Multimedia resources:

Laptop, geogebra file, projector and a pointer.

  • Website interactives/ links/ / Geogebra Applets

  • Process:
  1. The teacher can review the concept of a circle , radius , diameter and circumference .
  2. Any two points on the circumference can be joined.
  3. The joining line segment is called the chord.
  4. Let the students name the chord .
  5. Move the chord on the geogebra and let them observe its changing size.
  6. Let them observe the chord becoming a diameter while passing through the centre of the circle.
  7. The length of the chord is measued using a scale with its unit being cm.
  • Developmental Questions:
  1. The teacher can point to centre of circle and ask the students as to what it is.
  2. She can point to radius and ask the students to name it.
  3. Then ask them if any two points on the circumference are joined by a line segment what is it called ?
  4. How many chords can be drawn in a circle ?
  5. Are lengths of all chords the same ?
  6. Name the biggest chord in a circle.
  7. How do you measure a chord and in what units ?
  • Evaluation:

Were the students able to distinguish between radius, diameter and chord ?

  • Question Corner:

After drawing a chord,what are the two segregated parts of the circle called ?


Concept # 3 Angles in circles

Learning objectives

  1. students should understand that the angle at the centre of the circle is 360 degrees.

Notes for teachers

Activity No # 1.The angle at the centre is double the angle at the circumference

  • Estimated Time :40 minutes
  • Materials/ Resources needed :Laptop, geogebra file, projector and a pointer.
  • Prerequisites/Instructions, if any
  1. The students should have prior knowledge of a circle and circumference.
  2. They should know that an arc is a curved line along the circumference of a circle.
  3. If the end points of an arc are joined to a third point on the circumference of a circle, then an angle on the circumference is formed.
  4. If the end points of an arc are joined to the centre of a circle, then an angle at the centre of the circle is formed.
  5. They should know to measure the angles.
  • Multimedia resources: Laptop and a projector.
  • Website interactives/ links/ / Geogebra Applets

  • Process:
  1. The teacher should initially discuss about the circle , radius, centre and circumference.
  2. Projecting geogebra file she can show the major and the minor arcs.
  3. Name the arc in discussion.
  4. Let students find out and name the angle subtended by the arc at the centre and angle subtended by the same arc on the circumference.
  5. Observe that the end points of the arc lie on the angle.
  6. Each side of the angle contains at least one end -point of the arc.
  7. Project different angles subtended by the same arc on the circumference. What is the inference ?
  8. Compare angle formed at the centre and angle formed on the circumference by the same arc.
  9. Change the angles/arc using slider. Note down the two angles in each case.
  10. Ask students what they observed ? Let them infer.
  • Developmental Questions:
  1. Name the centre of the circle?
  2. Name the minor arc.
  3. Name the point on the circumference of the circle at which the arc subtends an angle.
  4. Name all radii from figure.
  5. What type of triangle is triangel APO ?
  6. Name the two equal sides of the triangle APO.
  7. Recall the theorem related to isosceles triangle.
  8. Name the two equal angles.
  9. Name the exterioe angle for the triangle APO
  10. Recall the exterior angle theorem.
  11. What relation do you observe between <p and <x.
  12. Similarly try to explain the same with triangle PBO.
  13. If <APO is half of <AOQ and <BPO is half of <BOQ what can you conclude about angles <AOB and <APB.
  14. What relation do you observe between the angle at the centre and that on the circumference formed by the same arc ?
  • Evaluation:
  1. In a circle, how many angles are subtended by an arc at its centre?
  2. In a circle, how many angles are subtended by an arc at its circumference?
  • Question Corner:
  1. What are the applications of this theorem.

Activity No # 2. Angles in a circle.

  • Estimated Time: 40 minutes
  • Materials/ Resources needed:Laptop, projector, geogebra file and a pointer.
  • Prerequisites/Instructions, if any
  1. The students should have prior knowledge of a circle, angles, arcs and segments.
  2. The students should have a thorough knowledge about the types of angles.
  3. They should have the skill of drawing a circle , angles and measuring them.
  • Multimedia resources : Laptop, Projector.
  • Website interactives/ links/ / Geogebra Applets

  • Process:
  1. The teacher can recall the concept of circle, arc segment.
  2. She can then project the geogebra file , change slider and make clear the theorems about angles in a circle.

Developmental Questions:

  1. Name the minor and major segments.
  2. Name the angles formed by them.
  3. Where are the two angles subtended ?
  4. What is the relation between the two angles.
  5. Name the major and minor arcs.
  6. What is an acute angle?
  7. What is an obtuse angle?
  8. What type of angles are formed by minor arc ?
  9. What type of angles are formed by major arc ?
  10. What are your conclusions ?
  • Evaluation:
  1. How many angles can a segment subtend on the circumference ?
  2. What can you say about these angles ?
  • Question Corner:
  1. Recall the theorems related to angles in a circle.

Concept # 4. Finding the Circumference of a circle

Learning objectives

  1. The children understand that the distance around the edge of a circle is known as circumference.
  2. The children learn to measure the circumference of the circle.
  3. Derivation of formula for circumference.
  4. They understand what is pi.

Notes for teachers

The circumference of a circle relates to one of the most important mathematical constants in all of mathematics. This constant pi, is represented by the Greek letter П. The numerical value of π is 3.14159 26535 89793 , and is defined by the ratio of a circle's circumference to its diameter. C = п. D or C = 2пr.

Activity No # 1 Derivation of formula for circumference and the value for pi.

  • Estimated Time : 45 mins
  • Materials/ Resources needed:

Note books, compass, pencil, mender, scale.

  • Prerequisites/Instructions, if any:
  1. The children should have prior knowledge of circle, radius, diameter and circumference.
  2. They should have measuring and computational skills.
  • Multimedia resources:
  • Website interactives/ links/ / Geogebra Applets
  • Process:
  1. Ask the children to draw five circles with different radii.
  2. Let them carefully measure their circumferences using wool.
  3. Mark the distance around the circle on the wool with a sketch pen.
  4. Measure the length of the measured wool using a scale.
  5. Make a table with columns radius, diameter and circumference
  6. For every circle find Circumference / diameter.
  7. Round C/d to two decimal places.
  8. Observe the answers in each case. It would be aprroximately 3.14 .
  9. The value 3.14 is the value of pi which is constant.

C/d = п or C = п d or C = 2п r.

  • Developmental Questions:
  1. Have you noted down radius, diameter and their respective circumferences.
  2. Check if your calculations are correct.
  3. What do you infer from the observed results ?
  • Evaluation:
  1. Are the children taking correct measurements.
  2. Are they comparing the results of C/d with all circles.
  3. Are they noticing that it is constant .
  4. Are they questioning why it is constant?
  • Question Corner:
  1. How do you derive the formula for circumference of a circle ?
  2. What is the name of that constant ?
  3. Try to collect more information on Pi.

Concept # 5 Finding the area of a circle.

Learning objectives

  1. The child should understand that the area of a circle is the entire planar surface.
  2. Derivation of the formula for area of the circle.
  3. Area of the circle is dependent on its radius.
  4. The formula for area of a circle is derived by converting the circle into an equally sized parallelogram.

Notes for teachers

1.Proof for area of a circle refer to them following link. http://www.basic-mathematics.com/proof-of-the-area-of-a-circle.html

Activity No # 1. To discover a formula for the area of a circle.

This activity has been taken from website : http://www.mathsteacher.com.au/year8/ch12_area/07_circle/circle.htm

  • Estimated Time:90 mins
  • Materials/ Resources needed:A compass, pair of scissors, ruler and protractor , pencil and chart papers.
  • Prerequisites/Instructions, if any
  1. Prior knowledge of circle, radius and parallelogram area.
  2. Skill to measure and draw accurately.
  • Multimedia resources
  • Website interactives/ links/ / Geogebra Applets
  • Process:

Refer this website : http://www.mathsteacher.com.au/year8/ch12_area/07_circle/circle.htm

  • Developmental Questions:
  1. Calculate the area of the figure in Step 6 by using the formula: Area = base x height
  2. What is the area of the circle drawn in Step 1?
  3. It appears that there is a formula for calculating the area of a circle. Can you discover it?
  • Evaluation:
  1. Is the student able to comprehend the idea of area.
  2. Is the student able to corelate that the base of the parallelogram formed is half of the circle's circumference.
  • Question Corner:
  1. What is the area of a parallelogram ?
  2. Is there any other way by which you can deduce the formula for area of a circle ?

Activity No # 2. Proving area of the circle = п r² using geogebra applet.

  • Estimated Time: 45mins
  • Materials/ Resources needed;

Laptop, geogebra file, projector and a pointer.

  • Prerequisites/Instructions, if any:

Prior knowledge of circle, radius, square and area of square.

  • Multimedia resources: Laptop.
  • Website interactives/ links/ / Geogebra Applets

  • Process:
  1. Show the students the two figures circle and square.
  2. Tell them that the radius and side of square are of same measure as it would help us in deducing the formula for area of circle.
  3. Formulas are easy ways of calculating area .
  4. If formulas are not known then the entire area in question can be divided into small squares of 1 unit measure and can deduce the formula of the whole.
  5. First the number of full squares is counted.
  6. Then two half squares would add up to 1 full square.
  7. Ignore less than quarter . Take 3/4 as full.
  8. Finally adding up the whole number would give us the full area of the figure in question.
  9. Divide area of circle with that of square and deduce formula for square with known formula for square.
  • Developmental Questions:
  1. Which are these two figures?
  2. What inputs do you need to draw a circle ? And for a square ?
  3. What do you observe as constant in the two figures ?
  4. Do you think the size of both the figures are same ?
  5. How do we find it ?
  6. What is the formula to find the area of a square ?
  7. When we do not know the formula for area, how do we deduce it ?
  8. Count the number of squares in the entire area of circle ?
  9. How to add half and quarter squares ?
  10. Approximately how many total 1 unit squares cover the circle ?
  11. So, what is the area of the circle ?
  12. What are we trying to deduce (get) through this activity ?
  13. Fine lets try dividing the area of circle with area of square and observe the proceedings while we try to deduce the formula for area of circle.
  • Evaluation;
  1. Has the student understood the concept of area.
  2. Was the student aligned with the assignment and was he able to follow the sequence of steps ?
  3. Is the student able to appreciate the analogy ?
  • Question Corner;
  1. What is Pi ?
  2. What do you understand by area ?
  3. What is the formula to find the area of square and that of a circle ?

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