CIRCLES - class 11 th

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Published on March 7, 2020 2:37:50 PM ● Video Link: https://www.youtube.com/watch?v=qPKjBamlpas

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Counter-Strike: Source (2004)

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A circle is a shape consisting of all points in a plane that are a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.

A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus.

Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.[3][4]

Some highlights in the history of the circle are:

1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to
256
/
81
(3.16049...) as an approximate value of π.[5]

Tughrul Tower from inside
300 BCE – Book 3 of Euclid's Elements deals with the properties of circles.
In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation.
1880 CE – Lindemann proves that π is transcendental, effectively settling the millennia-old problem of squaring the circle.[6]
Analytic results
Length of circumference
Further information: Circumference
The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:

{\displaystyle C=2\pi r=\pi d.\,}C=2\pi r=\pi d.\,
Area enclosed

Area enclosed by a circle = π × area of the shaded square
Main article: Area of a circle
As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius,[7] which comes to π multiplied by the radius squared:

{\displaystyle \mathrm {Area} =\pi r^{2}.\,}\mathrm {Area} =\pi r^{2}.\,
Equivalently, denoting diameter by d,

{\displaystyle \mathrm {Area} ={\frac {\pi d^{2}}{4}}\approx 0{.}7854d^{2},}\mathrm {Area} ={\frac {\pi d^{2}}{4}}\approx 0{.}7854d^{2},
that is, approximately 79% of the circumscribing square (whose side is of length d).

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.

Equations
Cartesian coordinates

Circle of radius r = 1, centre (a, b) = (1.2, −0.5)
Equation of a circle
In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that

{\displaystyle \left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}.}\left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}.
This equation, known as the Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |x − a| and |y − b|. If the circle is centred at the origin (0, 0), then the equation simplifies to

{\displaystyle x^{2}+y^{2}=r^{2}.\!\ }x^{2}+y^{2}=r^{2}.\!\
Parametric form
The equation can be written in parametric form using the trigonometric functions sine and cosine as

{\displaystyle x=a+r\,\cos t,\,}x=a+r\,\cos t,\,
{\displaystyle y=b+r\,\sin t\,}y = b+r\,\sin t\,
where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the positive x-axis.

An alternative parametrisation of the circle is:

{\displaystyle x=a+r{\frac {1-t^{2}}{1+t^{2}}}.\,}

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