# Measurements in circles

## Concept # Measurements in circles: 1. Radius and Diameter

### Learning objectives

- Ability to measure radius, diameter, circumference, chord length and angles subtended at the centre and on the circumference of the circle.
- Radius, diameter and chord lengths are linear measurements.
- Relate the size of the circle with radius.
- They realise that to draw a circle knowing the measure of radius or diameter is essential.
- There can be infinite radii in a circle.
- Diameter is twice the radius.
- The students should understand what a chord is.
- Chords of different lengths can be drawn in a circle.
- Chord length can be measured using a scale and its units is cm.
- The length of the chord increases as it moves closer to the diameter.
- The longest chord in the circle is its diameter.
- Distance of chord from the centre is its perpendicular distance from the centre.
- A chord divides the circle into two segments.
- Angle at the centre of the circle is 360º.
- Angles in circles are measured using protractor.
- Circumference and area are calculated using formula.

### Notes for teachers

### Activity No # 1. Measuring radius and diameter.

- Estimated Time: 15 mins
- Materials/ Resources needed:

- Laptop, goegebra tool, projector and a pointer.
- students' geometry box

- Prerequisites/Instructions, if any:

- Circle and its basic parts should have been done.

- Multimedia resources: Laptop
- Website interactives/ links/ / Geogebra Applets : This file was done by ITfC-Edu-Team.

- Process:

- Initially the teacher can explain the terms: circle, its centre, radius, diameter and circumference.
- Ask the children “What parameter is needed to draw a circle of required size ?”
- Show them how to measure radius on the scale accurately using compass.
- Show them to draw a circle.
- Given diameter, radius = D/2.
- 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.
- To measure 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:

- Name the centre of the circle.
- Name the point on the circumference of the circle.
- What is the linesegment AB called ?
- Name the line passing through the centre of the circle.
- Using what can you measure the radius and diameter.
- Name the units of radius/diameter.

- Evaluation:

- How do you measure exact radius on the compass?
- Are the children able to corelate the radius/diameter of a circle with its size ?

- Question Corner:

- If the centre of the circle is not marked , then how do you get the radius for a given circle.
- How many radii/diameter can be drawn in a circle?
- Are all radii for a given circle equal ?
- Is a circle unique for a given radius/diameter ?
- 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:
- Multimedia resources:

Laptop, geogebra file, projector and a pointer.

- Website interactives/ links/ / Geogebra Applets

- Process:

- Show the geogebra file and ask the following questions.

- Developmental Questions:

- The teacher can point to centre of circle and ask the students as to what it is.
- She can point to radius and ask the students to name it.
- Then ask them if any two points on the circumference are joined by a line segment what is it called ?
- How many chords can be drawn in a circle ?
- Are lengths of all chords the same ?
- Name the biggest chord in a circle.
- How do you measure a chord and in what units ?

- Evaluation:

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

- Question Corner:

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

## Concept # 2. Angles in circles

### Learning objectives

- 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

- Circles and its parts should have been done.

- Multimedia resources: Laptop and a projector.
- Website interactives/ links/ / Geogebra Applets

- Process:

- Project the geogebra file and ask the questions listed below.

- Developmental Questions:

- Name the centre of the circle?
- Name the minor arc.
- Name the point on the circumference of the circle at which the arc subtends an angle.
- Name all radii from figure.
- What type of triangle is triangel APO ?
- Name the two equal sides of the triangle APO.
- Recall the theorem related to isosceles triangle.
- Name the two equal angles.
- Name the exterioe angle for the triangle APO
- Recall the exterior angle theorem.
- What relation do you observe between <p and <x.
- Similarly try to explain the same with triangle PBO.
- If <APO is half of <AOQ and <BPO is half of <BOQ what can you conclude about angles <AOB and <APB.
- What relation do you observe between the angle at the centre and that on the circumference formed by the same arc ?

- Evaluation:

- In a circle, how many angles are subtended by an arc at its centre?
- In a circle, how many angles are subtended by an arc at its circumference?

- Question Corner:

- 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

- Knowledge of a circle, angles, arcs and segments.
- About the types of angles.
- Skill of drawing a circle , angles and measuring them.

- Multimedia resources : Laptop, Projector.
- Website interactives/ links/ / Geogebra Applets: This file has been done by Mallikarjun Sudi of Yadgir.

- Process:

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

Developmental Questions:

- Name the minor and major segments.
- Name the angles formed by them.
- Where are the two angles subtended ?
- What is the relation between the two angles.
- Name the major and minor arcs.
- What is an acute angle?
- What is an obtuse angle?
- What type of angles are formed by minor arc ?
- What type of angles are formed by major arc ?
- What are your conclusions ?

- Evaluation:

- How many angles can a segment subtend on the circumference ?
- What can you say about these angles ?

- Question Corner:

- Recall the theorems related to angles in a circle.

- Process:

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

Developmental Questions:

- Name the minor and major segments.
- Name the angles formed by them.
- Where are the two angles subtended ?
- What is the relation between the two angles.
- Name the major and minor arcs.
- What is an acute angle?
- What is an obtuse angle?
- What type of angles are formed by minor arc ?
- What type of angles are formed by major arc ?
- What are your conclusions ?

- Evaluation:

- How many angles can a segment subtend on the circumference ?
- What can you say about these angles ?

- Question Corner:

- Recall the theorems related to angles in a circle.

## Concept # 3. Finding the Circumference of a circle

### Learning objectives

- The children understand that the distance around the edge of a circle is known as circumference.
- The children learn to measure the circumference of the circle.
- Derivation of formula for circumference.
- 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:

- Circles basics should have been done.

- Multimedia resources:
- Website interactives/ links/ / Geogebra Applets
- Process:

- Ask the children to draw five circles with different radii.
- Let them carefully measure their circumferences using wool.
- Mark the distance around the circle on the wool with a sketch pen.
- Measure the length of the measured wool using a scale.
- Make a table with columns radius, diameter and circumference
- For every circle find Circumference / diameter.
- Round C/d to two decimal places.
- Observe the answers in each case. It would be aprroximately 3.14 .
- 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:

- Have you noted down radius, diameter and their respective circumferences.
- Check if your calculations are correct.
- What do you infer from the observed results ?

- Evaluation:

- Are the children taking correct measurements.
- Are they comparing the results of C/d with all circles.
- Are they noticing that it is constant .
- Are they questioning why it is constant?

- Question Corner:

- How do you derive the formula for circumference of a circle ?
- What is the name of that constant ?
- Try to collect more information on Pi.

## Concept # 4. Finding the area of a circle.

### Learning objectives

- The child should understand that the area of a circle is the entire planar surface.
- Derivation of the formula for area of the circle.
- Area of the circle is dependent on its radius.
- 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

- Prior knowledge of circle, radius and parallelogram area.
- 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:

- Calculate the area of the figure in Step 6 by using the formula: Area = base x height
- What is the area of the circle drawn in Step 1?
- It appears that there is a formula for calculating the area of a circle. Can you discover it?

- Evaluation:

- Is the student able to comprehend the idea of area.
- Is the student able to corelate that the base of the parallelogram formed is half of the circle's circumference.

- Question Corner:

- What is the area of a parallelogram ?
- 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: This file was done by Bindu.

- Process:

- Show the students the two figures circle and square.
- 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.
- Formulas are easy ways of calculating area .
- 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.
- First the number of full squares is counted.
- Then two half squares would add up to 1 full square.
- Ignore less than quarter . Take 3/4 as full.
- Finally adding up the whole number would give us the full area of the figure in question.
- Divide area of circle with that of square and deduce formula for square with known formula for square.

- Developmental Questions:

- Which are these two figures?
- What inputs do you need to draw a circle ? And for a square ?
- What do you observe as constant in the two figures ?
- Do you think the size of both the figures are same ?
- How do we find it ?
- What is the formula to find the area of a square ?
- When we do not know the formula for area, how do we deduce it ?
- Count the number of squares in the entire area of circle ?
- How to add half and quarter squares ?
- Approximately how many total 1 unit squares cover the circle ?
- So, what is the area of the circle ?
- What are we trying to deduce (get) through this activity ?
- 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;

- Has the student understood the concept of area.
- Was the student aligned with the assignment and was he able to follow the sequence of steps ?
- Is the student able to appreciate the analogy ?

- Question Corner;

- What is Pi ?
- What do you understand by area ?
- What is the formula to find the area of square and that of a circle ?

# Hints for difficult problems

# Project Ideas

# Math Fun

**Usage**

Create a new page and type {{subst:Math-Content}} to use this template