Topicmb58ff723d0d3056a_1528449000663_0Topic

The circle sector and the circle segment

Levelmb58ff723d0d3056a_1528449084556_0Level

Third

Core curriculummb58ff723d0d3056a_1528449076687_0Core curriculum

VIII. Plane geometry. The student:

6) uses formulae for the area of the circlearea of the circlearea of the circle sector and the length of the circle segmentcircle segmentcircle segment;

11) uses trigonometric functions to find the length of line segments in plane figures and calculate the areas of figures.

Timingmb58ff723d0d3056a_1528449068082_0Timing

45 minutes

General objectivemb58ff723d0d3056a_1528449523725_0General objective

Using mathematical objects and manipulating them, interpreting mathematical concepts.

Specific objectivesmb58ff723d0d3056a_1528449552113_0Specific objectives

1. Derivation of the formulae for the area of the circle sectorcircle sectorcircle sector and the area of the circlearea of the circlearea of the circle segment.

2. Calculating the area of the circlearea of the circlearea of the circle sector and the area of the circlearea of the circlearea of the circle segment.

3. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

Learning outcomesmb58ff723d0d3056a_1528450430307_0Learning outcomes

The student:

- derives the formulae for the area of the circlearea of the circlearea of the circle sector and the area of the circlearea of the circlearea of the circle segment,

- calculates the area of the circlearea of the circlearea of the circle sector and the area of the circlearea of the circlearea of the circle segment.

Methodsmb58ff723d0d3056a_1528449534267_0Methods

1. Discussion.

2. Situational analysis.

Forms of workmb58ff723d0d3056a_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmb58ff723d0d3056a_1528450127855_0Introduction

The teacher informs the students that during the class they will learn to calculate the area of the circlearea of the circlearea of the circle sector and the area of the circlearea of the circlearea of the circle segment. 

The students recollect the definition of the central anglecentral anglecentral angle. They recollect the formula for the area of the circlearea of the circlearea of the circle and the formula for the areaformula for the areaformula for the area of the triangle when two of its sides and the angle between them is known.

Proceduremb58ff723d0d3056a_1528446435040_0Procedure

The teacher gives the definition of the circle sectorcircle sectorcircle sector.

Definition – the circle sectorcircle sectorcircle sector.

- The circle sectorcircle sectorcircle sector is the part of the disk enclosed by the arms of the central anglecentral anglecentral angle $\alpha$ and the subtended arc.

[Illsutration 1]

The teacher gives the definition of the circle segmentcircle segmentcircle segment.

The definition - the circle segment.

- The circle segmentcircle segmentcircle segment is the part of the disk enclosed by the chordchordchord, indicating the central angle $\alpha$ and the subtended arc.

[Illustraion 2]

The students work individually or in pairs, using their computers. Their task is to observe the relation between the ratio of the area of the circlearea of the circlearea of the circle sector to the area of the circle and the ratio of the central anglecentral anglecentral angle to the full angle.

Task 
Open the Geogebra: „The circle sectorcircle sectorcircle sector” applet. Change the position of point A and answer the questions:

1. How does the ratio of the area of the circlearea of the circlearea of the circle sector to the area of the circle change?
2. How does the ratio of the central anglecentral anglecentral angle to the full angle change?
3. Is the area of the circlearea of the circlearea of the circle sector proportional to the area of the circle? Does the constant of proportionality depend on the measure of the angle of the circle sectorcircle sectorcircle sector?

[Geogebra applet]

Suggest a formula for the area of the circlearea of the circlearea of the circle sector based on your observations.

The students cooperate to decide what the formula for the area of the circlearea of the circlearea of the circle sector is.

The formula for the area of the circlearea of the circlearea of the circle sector.

- The area of the circle sector with radius $r$ cut out by the central angle measuring $\alpha$ is expressed with formula:mb58ff723d0d3056a_1527752263647_0- The area of the circle sector with radius $r$ cut out by the central angle measuring $\alpha$ is expressed with formula:

Discussion
How can you calculate the circle segmentcircle segmentcircle segment?

The students look at the diagram and notice that the area of the segment is the difference between the area of the circlearea of the circlearea of the circle sector with central anglecentral anglecentral angle $\alpha$ and the area of triangle $\mathrm{ABS}$. Triangle $\mathrm{ABS}$ is isosceles with sides being the radii of the circle and angle $\alpha$ between them. The area of triangle $\mathrm{ABS}$ equals $P_{ABS}=\frac{1}{2}·r^2·sin\alpha$. So, the area of the segment may be derives form the formula:

Then, the students check the correctness of the formula for $\alpha =180°$ and for $\alpha >180°$.

The students solve the task individually and later, discuss it with the whole class.

Task 
1. In the circle with radius of 5 cm there is a chordchordchord of 5 cm. Calculate the area of the circle segmentcircle segmentcircle segment.

2. Inscribed angle measures 60° and intercepts the arc AB of circle whose radius is 3 cm. Calculate the area of the sector of the circle enclosed by arc AB.  

3. The area of the circlearea of the circlearea of the circle is $144\pi$ cmIndeks górny 2². Find the measure of the central anglecentral anglecentral angle, for which the area of the sector of this circle equals $36\pi$ cmIndeks górny 2².

An extra task:
A circle with radius of 1 cm was inscribed in the circle sector, as show in the diagram. Calculate the area of this sector.mb58ff723d0d3056a_1527752256679_0An extra task:
A circle with radius of 1 cm was inscribed in the circle sector, as show in the diagram. Calculate the area of this sector.

[Illsutration 3]

Lesson summarymb58ff723d0d3056a_1528450119332_0Lesson summary

The students do the consolidation tasks.

Then, they summarize the class.

The area of the circlearea of the circlearea of the circle sector with radius $\mathrm{r}$ cut out by the central anglecentral anglecentral angle measuring $\alpha$ is expressed with formula:

The area of the segment of the circle whose radius is $r$ and the central anglecentral anglecentral angle $\alpha$ is expressed with formula:

Selected words and expressions used in the lesson plan

area of the circlearea of the circlearea of the circle

central anglecentral anglecentral angle

chordchordchord

circle radiuscircle radiuscircle radius

circle sectorcircle sectorcircle sector

circle segmentcircle segmentcircle segment

formula for the areaformula for the areaformula for the area

subtended by the arcsubtended by the arcsubtended by the arc