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Circle - Mathematical definitions

Circle - Mathematical definitions: Encyclopedia II - Circle - Mathematical definitions

In an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that If the circle is centred at the origin (0, 0), then this formula can be simplified to: The circle centred at the origin with radius 1 is called the unit circle. Expressed in parametric equations, (x, ySee also:

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Circle, Circle - Chord properties, Circle - Inscribed angle theorem, Circle - Mathematical definitions, Circle - Properties, Circle - Secant tangent and chord properties, Circle - Tangent properties, Sphere, Unit circle, Descartes' theorem, Isoperimetric theorem, List of circle topics

Circle: Encyclopedia II - Circle - Mathematical definitions

Circle - Mathematical definitions

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

If the circle is centred at the origin (0, 0), then this formula can be simplified to:

The circle centred at the origin with radius 1 is called the unit circle.

Expressed in parametric equations, (x, y) can be written as

The slope a circle at a point (x, y) can be expressed with the following formula, assuming the centre is at the origin and (x, y) is on the circle:

In the complex plane, a circle with a centre at c and radius r has the equation . Since , the slightly generalized equation for real p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles.

All circles are similar; as a consequence, a circle's circumference and radius are proportional, as are its area and the square of its radius. The constants of proportionality are 2π and π, respectively. In other words:

Length of a circle's circumference =
Area of a circle =

The formula for the area of a circle can be derived from the formula for the circumference and the formula for the area of a triangle, as follows. Imagine a regular hexagon (six-sided figure) divided into equal triangles, with their apices at the centre of the hexagon. The area of the hexagon may be found by the formula for triangle area by adding up the lengths of all the triangle bases (on the exterior of the hexagon), multiplying by the height of the triangles (distance from the middle of the base to the center) and dividing by two. This is an approximation of the area of a circle. Then imagine the same exercise with an octagon (eight-sided figure), and the approximation is a little closer to the area of a circle. As a regular polygon with more and more sides is divided into triangles and the area calculated from this, the area becomes closer and closer to the area of a circle. In the limit, the sum of the bases approaches the circumference 2πr, and the triangles' height approaches the radius r. Multiplying the circumference and radius and dividing by 2, we get the area, π r².

Other related archives

Descartes' theorem, Euclidean geometry, Isoperimetric theorem, List of circle topics, Power of a point, Sphere, Unit circle, approximation, area, complex plane, cone, conic section, constants, coordinate system, disk, distance, formula, hexagon, limit, octagon, parametric equations, plane, points, proportional, set, similar, simple closed curves, slope, triangle, unit circle, π

Adapted from the Wikipedia article "Mathematical definitions", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki

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