Areas Related to Circles in Mathematics
The concepts of circumference, area of circles, major/minor segments, sectors, and their applications in real-life situations. Understand how to find areas and lengths related to circular figures. Delve into practical examples and calculations in this comprehensive study.
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Presentation Transcript
MATHEMATICS STD-X TOPIC: AREAS RELATED TO CIRCLES
PDF CHAPTER LINK NCERT TEXTBOOK (CHAPTER-Area related to circle) NCERT EXEMPLAR(CHAPTER-Area related to circle)
Learning Objectives Students will be able to 1. recall the concept of circumference and area of circle and its use in daily life situations. 2. understand the terms major segments, minor segments, major sector, minor sector and angle subtended by the sector at the centre. 3. find area of a sector of circle of a given angle , length of an arc of a sector and their application. 4. apply the knowledge of area of plane figure in solving the problems with combination of plane figures in real life situations.
Introduction You are already familiar with perimeters and areas of simple plane figures such as rectangles, squares, parallelograms, triangles and circles from your earlier classes. Many objects that we come across in our daily life are related to the circular shape in some form or the other. Cycle wheels, round cake, circular clocks, DVD disc, papad, bangles, etc are some examples of such objects .
Daily life examples Cycle wheel is circular in shape Circular cake
Daily life examples Round clocks CD/DVD discs So, finding perimeters and areas related to circular figures is of great practical importance.
Perimeter Perimeter of of a Circle a Circle The distanced covered by travelling around a circle is its perimeter , usually called its circumference ????????????? ???????? = ? ????????????? = ? ???????? ????????????? = ? 2? (As diameter=2r) ????????????? = 2??
Area of a Circle Recall that you have verified it in Class VII, by cutting a circle into a number of sectors and rearranging them as shown you will get a rectangular shape. ???? ?? ?????? = ?? ? = ??2
Example 1 Question: The cost of fencing a circular field at the rate of Rs 24 per metre is Rs 5280. The field is to be ploughed at the rate of Rs 0.50 per ?2. Find the cost of ploughing the field. Solution: Length of the fence (in metres) (3 marks) =????? ???? ???? =5280 = 220? 24 ( mark)
Cont circumference of the field = 220 m 2?? = 220 ? =220 7 2 22 = 35? (1mark) Area of the field =??2=22 7 35 35 = 22 5 35?2 (1mark) Cost of ploughing the field=22 5 35 0.5 = ??1925 ( mark)
Sector of a circle The portion (or part) of the circular region enclosed by two radii and the corresponding arc is called a sector of the circle. The region enclosed by radii and minor arc is called minor sector The region enclosed by radii and major arc is called major sector
Activity 1 Objective: To find the area of a sector of a circle for given sector angle and radius of the circle Materials Required : Coloured paper, pair of scissor, geometry box, sketch pens. Procedure: Cut three circles of same radius from coloured sheet of paper. Let the radius be r units. In 1st circle shade half of the circular region by dividing it in to two equal parts.
Cont In the 2nd circle shade th of the circular region by dividing in to four equal parts. In the 3rd circle draw a sector in the circle for some sector angle and shade it. Let the sector angle in this case be ?. See the video of the activity video 2nd Circle 1st Circle 3rd Circle
Cont Observation: Here we observed that 1. In the first circle area of shaded part = half of the area of the circle =1 sector angle) 2. In the 2nd circle area of shaded part = One fourth of the area of the circle = 1 2??2=180 360 ??2 (where 1800 is 90 360 ??2 (where 900 is sector angle) 4??2=
Cont 3. In all circles the shaded part is nothing but sector of a circle. 4. We have seen that in 1st and 2nd circle area of shaded =area of the sector =?????? ????? 360 So in third circle area of shaded part = area of sector of sector angle ? = ??2 ? 360 ??2 Conclusion: From the above activity we conclude that area of sector of a circle is of the circle. ? 360 ??2. Where ?is the sector angle and r is the radius
Art Integration Figure 1 Figure 2
Area of sector of a circle When degree measure of the angle at the center is 360, area of the sector = ??2 So, when the degree measure of the angle at the Centre is 1, area of the sector =??2 Therefore, when the degree measure of the angle at the center is ?, 360 ? ??? ??? Area of the sector =
Length of an arc of a circle Length of the circle (of angle 360 ) =2?? Length of an arc of a sector of angle ? ? = 360 2??
Area of a segment of a circle Area of the minor segment APB = Area of the minor sector OAPB Area of OAB Area of the major segment AQB = ??2 area of minor segment
Example 2 Question: Find the area of the sector of a circle with radius 4 cm and of angle 30 . Also, find the area of the corresponding major sector. Solution: Area of the sector OAPB= (3 Marks) ? 360 ??2 3600 22 300 =4.19??2 Area of the corresponding major sector = ??2 area of sector OAPB = (3.14 16 4.19) = 46.05 = 46.1 ??2(approx.) = 7 4 4 (1 mark) (1 mark)
Example 3 Question: Find the area of the segment AYB shown in Figure, if radius of the circle is 21 cm and AOB = 120 . Solution: Area of the segment AYB = Area of sector OAYB Area of OAB Now, area of the sector OAYB = 120 =462 ??2 (4 Marks) 360 22 7 21 21 (1mark)
Cont For finding the area of OAB, draw OM AB as shown in figure Note that OA = OB. Therefore, by RHS congruence, AMO BMO. M is the mid-point of AB and AOM = BOM In ???, ???600=?? 21 3 2 So AB=2 ?? = 21 3 ?? Again ???600=?? =600 ( mark) =?? 21, ??,?? =21 3 2 ( mark) 21, ?? ?? =21 2?? ( mark)
Cont Area of ??? =1 2 ?? ?? 2 21 3 21 =1 Therefore, area of the segment AYB = 462 441 3 =21 4 2 =441 3 ( mark) ??2 4 4 (1mark) 88 21 3 ??2
Areas of Combinations of Plane Figures Example 4: Question: Find the area of the shaded region in figure, where ABCD is a square of side 14 cm Solution: Area of square ABCD = 14 14 cm2 = 196??2 (1mark) Diameter of each circle = 7cm, radius = 7 Area of one circle = ??2=22 Therefore, area of the four circles = 4 77 Hence, area of the shaded region = (196 154)=42??2 (3 Marks) 2?? ( mark) 2=77 2= 154??2 7 7 2 7 2??2 (1mark) ( mark)
Example 5 Question: In figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the (3 Marks) (i) quadrant OACB (ii) shaded region. Solution: Given radius OB=OA=3.5cm , OD=2cm (i) Area of quadrant OACB= 1 =1 (ii) Area of shaded region= area of quadrant area( ???) = 9.625 1 = 6.125??2 4??2 4 22 7 3.5 3.5 =9.625??2 (1 mark) 2 3.5 2 (1 mark)
Example 6 Question: A calf is tied with a rope of length 6 m at the corner of a square grassy lawn of side 20 m. If the length of the rope is increased by 5.5m, find the increase in area of the grassy lawn in which the calf can graze. Solution: Let the calf be tied at the corner A of the square lawn So, required increase in area = 900 =? =75.625 ?2 ( mark) (2 Marks) 900 3600 ?62 (1 mark) 3600 ? 11.52 411.5 + 6 11.5 6
Summary In this chapter, we have studied the following points : 1. Circumference of a circle = 2?? 2. Area of a circle = ??2 3.Length of an arc of a sector of a circle with radius r and angle with degree measure ? is = ? 360 2?? 4.Area of a sector of a circle with radius r and angle with degree measure ? is = ? 360 ??2 5. Area of segment of a circle = Area of the corresponding sector Area of the corresponding triangle.
Learning outcomes 1. Students learn that length of an arc of a sector of a circle with radius r and angle with degree measure ? is = ? 360 2?? 2. Students learn that area of a sector of a circle with radius r and angle with degree measure ? is = ??2 3. Students learn to find area of a particular region in combination of plane figures. 4. Students learn how to apply the knowledge of area of plane figure in solving the problems with combination of plane figures in real life situations. ? 360
