Measurements in circles

flash

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.

Activity No # 1. Measuring radius and diameter.
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /> Then she can project the digital tool 'geogebra.' and further clarify concepts.
 * 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.


 * 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.
Laptop, geogebra file, projector and a pointer. Laptop, geogebra file, projector and a pointer. <ggb_applet width="1278" height="571" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> Were the students able to distinguish between radius, diameter and chord ? After drawing a chord,what are the two segregated parts of the circle called ?
 * Estimated Time : 10 minutes
 * Materials/ Resources needed:
 * 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:
 * 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:
 * Question Corner:

Learning objectives

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

Activity No # 1.The angle at the centre is double the angle at the circumference
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" />
 * 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.
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> Developmental Questions: Developmental Questions:
 * 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.
 * 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.
 * 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.
 * 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.

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.
Note books, compass, pencil, mender, scale. C/d = п   or    C = п d     or   C = 2п r.
 * Estimated Time : 45 mins
 * Materials/ Resources needed:
 * 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.
 * 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.

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 Refer this 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:
 * 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.
Laptop, geogebra file, projector and a pointer. Prior knowledge of circle, radius, square and area of square. <ggb_applet width="1280" height="600" version="4.0" 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 * Estimated Time: 45mins
 * Materials/ Resources needed;
 * Prerequisites/Instructions, if any:
 * 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 ?
 * So, what is the area of the circle ?
 * 1) What are we trying to deduce (get) through this activity ?
 * 2) 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 ?

= Hints for difficult problems =

= Project Ideas =

= Math Fun =

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