CIRCLE FORMULA AND THEORY DETAIL

A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius.

A circle is a simple closed curve which 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 the former and the latter is called a disk.

A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant.

A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone.

Terminology

Chord: a line segment whose endpoints lie on the circle.

Diameter: the longest chord, a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a segment, which is the largest distance between any two points on the circle.

Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.

Circumference: the length of one circuit along the circle itself.

Tangent: a straight line that touches the circle at a single point.

Secant: an extended chord, a straight line cutting the circle at two points.

Arc: any connected part of the circle's circumference.

Sector: a region bounded by two radii and an arc lying between the radii.

Segment: a region bounded by a chord and an arc lying between the chord's endpoints.

Chord, secant, tangent, radius, and diameter

Arc, sector, and segment

Analytic results

Length of 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:

$C = 2\pi r = \pi d.\,$

Area enclosed-

As proved by Archimedes, 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,[3] which comes to π multiplied by the radius squared:

$\mathrm{Area} = \pi r^2.\,$

Equivalently, denoting diameter by d

$\mathrm{Area} = \frac{\pi d^2}{4} \approx 0{.}7854d^2,$

Equations---

Cartesian coordinates--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

$\left(x - a \right)^2 + \left( y - b \right)^2=r^2.$

This equation, also known as Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, 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

$x^2 + y^2 = r^2.\!\$

The equation can be written in parametric form using the trigonometric functions sine and cosine as

$x = a+r\,\cos t,\,$

$y = b+r\,\sin t\,$

In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the centre parallel to the x-axis.In homogeneous coordinates each conic section with equation of a circle is of the form

$ax^2+ay^2+2b_1xz+2b_2yz+cz^2 = 0.\,$

Polar coordinates-

In polar coordinates the equation of a circle is:

$r^2 - 2 r r_0 \cos(\theta - \phi) + r_0^2 = a^2\,$

where a is the radius of the circle, is the polar coordinate of a generic point on the circle, and is the polar coordinate of the centre of the circle (i.e., r0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle). For a circle centred at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on the circle, the equation becomes

$r = 2 a\cos(\theta - \phi).\,$

In the general case, the equation can be solved for r, giving

$r = r_0 \cos(\theta - \phi) + \sqrt{a^2 - r_0^2 \sin^2(\theta - \phi)},$

the solution with a minus sign in front of the square root giving the same curve.

Complex plane--In the complex plane, a circle with a centre at c and radius (r) has the equation . In parametric form this can be written .

The slightly generalised equation for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with , since . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.

$|z-c|^2 = r^2\,$ ;

$z = re^{it}+c$ .

Properties-

The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.)

The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.

All circles are similar.

A circle's circumference and radius are proportional.

The area enclosed and the square of its radius are proportional.

The constants of proportionality are 2π and π, respectively.

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

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 Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

Chord

Chords are equidistant from the centre of a circle if and only if they are equal in length.

The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:

A perpendicular line from the centre of a circle bisects the chord.

The line segment (circular segment) through the centre bisecting a chord is perpendicular to the chord.

If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.

If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.

If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplemental.

For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

An inscribed angle subtended by a diameter is a right angle (see Thales' theorem).

The diameter is the longest chord of the circle.

If the intersection of any two chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then ab = cd.

If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then a2 + b2 + c2 + d2 equals the square of the diameter.

The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two chords intersecting at the same point, and is given by 8r 2 – 4p 2 (where r is the circle's radius and p is the distance from the center point to the point of intersection).

The disance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.

Tangent--

The line perpendicular drawn to a radius through the end point of the radius is a tangent to the circle.

A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle.

Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.

If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠BOA and ∠BPA are supplementary.

If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = 1⁄2arc(AQ).

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