Measurements in circles

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Textbook

ncert books

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

Concept # 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

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