Circle

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CIRCLE LINES

Circle slices

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Shatir500

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God the Geometer

Circle

A circle is a simple closed shape in Euclidean geometry. It is the set of all points in a plane that are at 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 of the points 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.

Definition

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 terms, however, the circle is only the boundary and the whole figure is called a disk.

A circle can be defined as the locus of all points that satisfy the equations

x^2 + y^2 = r^2

where (x, y) are the coordinates of any point on the circle, and r is the radius of the circle.

Properties

A circle has several unique properties that distinguish it from other shapes. It has the highest symmetry of any shape, being invariant under rotation, reflection, and translation, assuming the sphere is not considered in two dimensions. A circle's diameter, the longest distance between any two points on the circle, is twice the radius. The circumference, the distance around the circle, is π times the diameter. The area enclosed by a circle is π times the square of the radius.

Circles in Coordinate Systems

In the Cartesian coordinate system, a circle with a center at (a, b) and radius r can be described by the equation:

(x - a)^2 + (y - b)^2 = r^2

In the polar coordinate system, a circle with a center at the origin and radius r can be described by the equation:

r(θ) = r

where θ is the angle parameter.

Circle Geometry

Circle geometry involves the study of properties and applications of circles. It includes topics such as the angles formed by intersecting lines and circles, the properties of chords, tangents, and secants, and the theorems relating to these. The Pythagorean theorem is a fundamental principle in circle geometry, relating the radius of a circle to the hypotenuse of a right-angled triangle.

Applications

Circles have a wide range of applications in both the natural and applied sciences. In astronomy, the orbits of planets are often approximated as circles. In engineering and construction, circles are used in the design of structures, machinery, and systems requiring rotational symmetry. Circles also play a crucial role in the visual arts, serving as a fundamental element in design and composition.

See Also

• Sphere
• Ellipse
• Cylinder
• Cone