Embed Size (px)
Transcript of Digital sherin
- 1. HistoryThe compass in this 13th-century manuscript is a symbol of God's act of Creation.Notice also the circular shape of the haloThe word "circle" derives from the Greek (kirkos), itself a metathesis ofthe Homeric Greek (krikos), meaning "hoop" or "ring". The origins of the words"circus" and "circuit" are closely related.Circular piece of silk with Mongol images
2. Circles in an old Arabic astronomical drawing.The circle has been known since before the beginning of recorded history. Natural circleswould have been observed, such as the Moon, Sun, and a short plant stalk blowing in the windon sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, withrelated inventions such as gears, makes much of modern machinery possible. In mathematics, thestudy of the circle has helped inspire the development of geometry, astronomy, and calculus.DEFINITION OF CIRCLESA circle is a simple shape of Euclidean geometry that is the set of all points in a planethat are at a given distance from a given point, the centre. The distance between any of the pointsand the centre is called the radius. It can also be defined as the locus of a point equidistant from afixed point.A circle is a simple closed curve which divides the plane into two regions: an interior andan exterior. In everyday use, the term "circle" may be used interchangeably to refer to either theboundary of the figure, or to the whole figure including its interior; in strict technical usage, thecircle is the former and the latter is called a disk. 3. A circle can be defined as the curve traced out by a point that moves so that its distancefrom a given point is constant.A circle may also be defined as a special ellipse in which the two foci are coincident andthe eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter,using calculus of variationsProperties The circle is the shape with the largest area for a given length of perimeter. (Isoperimetricinequality.) The circle is a highly symmetric shape: every line through the centre forms a line ofreflection symmetry and it has rotational symmetry around the centre for every angle. Itssymmetry group is the orthogonal group O(2,R). The group of rotations alone is the circlegroup T. 4. All circles are similar.o A circle's circumference and radius are proportional.o The area enclosed and the square of its radius are proportional.o The constants of proportionality are 2 and , respectively. The circle which is centred at the origin with radius 1 is called the unit circle.o Thought of as a great circle of the unit sphere, it becomes the Riemannian circle. Through any three points, not all on the same line, there lies a unique circle. In Cartesiancoordinates, it is possible to give explicit formulae for the coordinates of the centre of thecircle and the radius in terms of the coordinates of the three given points.Terminology Arc: any connected part of the circle. Centre: the point equidistant from the points on the circle. Chord: a line segment whose endpoints lie on the circle. Circular sector: a region bounded by two radii and an arc lying between the radii. Circular segment: a region, not containing the centre, bounded by a chord and an arclying between the chord's endpoints. Circumference: the length of one circuit along the circle. Diameter: a line segment whose endpoints lie on the circle and which passes through thecentre; or the length of such a line segment, which is the largest distance between anytwo points on the circle. It is a special case of a chord, namely the longest chord, and it istwice the radius. Passant: a coplanar straight line that does not touch the circle. 5. Radius: a line segment joining the centre of the circle to any point on the circle itself; orthe length of such a segment, which is half a diameter. Secant: an extended chord, a coplanar straight line cutting the circle at two points. Semicircle: a region bounded by a diameter and an arc lying between the diameter'sendpoints. It is a special case of a circular segment, namely the largest one. Tangent: a coplanar straight line that touches the circle at a single point.Mathematical Constant Chord, secant, tangent, radius, and diameterArc, sector, and segment 6. CIRCUMFERENCE OF A CIRCLECircumference (from Latin circumferentia, meaning "to carry around") is the lineardistance around the edge of a closed curve or circular object. The circumference of a circle isof special importance in geometry and trigonometry. However "circumference" may also refer tothe edge of elliptical closed curve. Circumference is a special case of perimeter in that theperimeter is typically around a polygon while circumference is around a closed curve.Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre ororigin (O) in magenta. Circumference = diameter = 2 radius.The circumference of a circle is the distance around it. The term is used when measuringphysical objects, as well as when considering abstract geometric forms. 7. When a circle's radius is 1, its circumference is 2.When a circle's diameter is 1, its circumference is .Relationship with PiThe circumference of a circle relates to one of the most important mathematical constantsin all of mathematics. This constant, pi, is represented by the Greek letter . The numerical valueof is 3.14159 26535 89793 ... , and is defined by two proportionality constants. The firstconstant is the ratio of a circle's circumference to its diameter and equals . While the secondconstant is the ratio of the diameter and two times the radius and is used as to convert thediameter to radius in the same ratio as the first, . Both proportionality constants combine inrespect with circumference c, diameter d, and radius r to become: 8. The use of the mathematical constant is ubiquitous in mathematics, engineering, and science.While the constant ratio of circumference to radius also has many uses inmathematics, engineering, and science, it is not formally named. These uses include but are notlimited to radians, computer programming, and physical constants.Example-1What is the circumference of the circle?Circumference=2 .radiusC=2 3.14 3=6 3.14=18.84cmExample23cm 9. What is the circumference of a circle with a radius 5cm?Circumference=2 .radiusC=2 3.14 5=10 3.14=31.4cmExample3A circles circumference is 102 inches. What is the diameter of the circle?Circumference= .diameter102= .diameter102/ = .diameter/ 32.5=diameterEXERCISE1. What is the circles circumference? 10. 7cm2. What is the circumference of the circle pictured below?22cm3. A circles circumference is 22 inches. what is the radius of the circle?4. What is the circumference of the circle pictured below?12cm 11. PerimeterPerimeter is the distance around a two dimensional shape, or the measurement of the distancearound something; the length of the boundary.A perimeter is a path that surrounds a two-dimensional shape. The word comes from the Greekperi (around) and meter (measure). The term may be used either for the path or its length - it can bethought of as the length of the outline of a shape. The perimeter of a circle is called its circumference.Perimeter of a Circlewhere I s the radius of the circle and is the diameter.AREA OF CIRCLEThe area of a circle is all the space inside a circles circumferenceIn the picture on the above, the area of the circle isthe part of the circle that is red color.A circle has radius r ,the area of a circle is 12. Area = r2r = radiusArea = r2r = radius 13. The area of a circle is all the space inside a circles circumference.In the picture on the above, the area of the circle is the part of thecircle that is red color.Example-4What is the area of the circle?3cm 14. Area= r2=3.14 3 3=9 3.14=28.26cm2Example-5What is the diameter of a circle if its area is 360cm2?Area= r2360=3.14 r2r2 =360/3.14=114.591559r= 114.591559=10.7047447Diameter=2r=210.7047447=21.40948 15. EXERCISE1. What is this circles area?16cm2. What is the area of a circle with a radius 7 centimeters?3. What is the radius of a circle .if its area is 120 cm2?4. A circle has a diameter of 12 inches .What is its area in terms of ?