Maths........

A circle is one of the simple shapes of Euclidean geometry. It is the locus of all points in a plane at a constant distance, called the radius, from a fixed point, called the center. Through any three points not on the same line, there passes one and only one circle.

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Circle illustration showing a radius, a diameter, the center and the circumference.

Analytic results


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Circle of radius r=1, center (a, b)=(1.2, -0.5).

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In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that



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The equation of the circle follows from the Pythagorean theorem applied to any point on the circle. If the circle is centred at the origin (0, 0), then this formula can be simplified to


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When expressed in parametric equations, (x, y) can be written using the trigonometric functions sine and cosine as


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where t is a parametric variable, understood as the angle the ray to (x, y) makes with the x-axis. Alternatively, in stereographic coordinates, the circle has a parametrization


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In homogeneous coordinates each conic section with equation of a circle is


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It can be proven that a conic section is a circle if and only if the point I(1: i: 0) and J(1: −i: 0) lie on the conic section. These points are called the circular points at infinity.
Pi or π is the ratio of a circle's circumference to its diameter.
The numeric value of π never changes.
In modern English, it is pronounced /ˈpaɪ/ (as in apple pie).