Circle

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Circle illustration

This article is about the shape and mathematical concept of circle. For other uses, see Circle (disambiguation).

In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.

Circles are simple closed curves which divide the plane into an interior and exterior. The circumference of a circle means the length of the circle, and the interior of the circle is called a disk. An arc is any continuous portion of a circle.

A circle is a special ellipse in which the two foci coincide (i.e., are the same point). Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

Analytic results

Chord, secant, and tangent

Arc, sector, and segment

Equation of a circle

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

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

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

$x^{2}+y^{2}=r^{2}\!\$

and its tangent will be

$xx_{1}+yy_{1}=r^{2}\!\$

where $x_{1}$, $y_{1}$ the coordinates of the common point.

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

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

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

where t is a parametric variable, understood as the angle the ray to (x, y) makes with the x-axis.

In homogeneous coordinates each conic section with equation

$ax^{2}+ay^{2}+2b_{1}xz+2b_{2}yz+cz^{2}=0$

is called a circle.

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.

In polar coordinates the equation of a circle is:

$r^{2}-2rr_{0}\cos(\theta -\varphi )+r_{0}^{2}=a^{2}\,$

In the complex plane, a circle with a centre at c and radius r has the equation $|z-c|^{2}=r^{2}$. Since $|z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}$, the slightly generalized equation $pz{\overline {z}}+gz+{\overline {gz}}=q$ 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.

Slope

The slope of 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:

$y'=-{\frac {x}{y}}$

Area enclosed

Area of the circle = π × area of the shaded square

Main article: Area of a disk

• The area enclosed by a circle is

$A=r^{2}\cdot \pi ={\frac {d^{2}\cdot \pi }{4}}\approx 0{.}7854\cdot d^{2}$

that is approximately 79% of the circumscribing square.

Circumference

Main article: circumference

• Length of a circle's circumference is

$c=\pi d=2\pi \cdot r$

• Alternate formula for circumference:

Given that the ratio circumference c to the Area A is

${\frac {c}{A}}={\frac {2\pi r}{\pi r^{2}}}$

The $r$ and the $\pi$ can be canceled, leaving

${\frac {c}{A}}={\frac {2}{r}}$

Therefore solving for c:

$c={\frac {2A}{r}}$

So the circumference is equal to 2 times the area, divided by the radius. This can be used to calculate the circumference when a value for pi cannot be computed.

Diameter

Main article: diameter

The diameter of a circle is:

$d=2r=2\cdot {\sqrt {\frac {A}{\pi }}}\approx 1{.}1284\cdot {\sqrt {A}}$

Properties

Secant-secant theorem

• The circle is the shape with the highest area for a given length of perimeter. (See Isoperimetry)
• 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 centred at the origin with radius 1 is called the unit circle.

Chord properties

• Chords equidistant from the centre of a circle are equal (length).
• Equal (length) chords are equidistant from the centre.
• 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.

Tangent properties

• The line drawn perpendicular to the end point of a radius is a tangent to the circle.
• A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
• Tangents drawn from a point outside the circle are equal in length.
• Two tangents can always be drawn from a point outside of the circle.

Theorems

See also: Power of a point

• The chord theorem states that if two chords, CD and EF, intersect at G, then $CG\times DG=EG\times FG$. (Chord theorem)
• If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then $DC^{2}=DG\times DE$. (tangent-secant theorem)
• If two secants, DG and DE, also cut the circle at H and F respectively, then $DH\times DG=DF\times DE$. (Corollary of the tangent-secant theorem)
• The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent chord property)
• If the angle subtended by the chord at the centre is 90 degrees then l = √(2) × r, where l is the length of the chord and r is the radius of the circle.
• If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem.

Inscribed angles

Inscribed angle theorem

An inscribed angle $\psi$ is exactly half of the corresponding central angle $\theta$ (see Figure). Hence, all inscribed angles that subtend the same arc have the same value (cf. the blue and green angles $\psi$ in the Figure). Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle.

An alternative definition of a circle

${\frac {d_{1}}{d_{2}}}={\textrm {constant}}$ Apollonius' definition of a circle

Apollonius of Perga showed that a circle may also be defined as the set of points having a constant ratio of distances to two foci, A and B.

The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:

${\frac {AP}{BP}}={\frac {AC}{BC}}$

Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to $180^{\circ }$, the angle CPD is exactly $90^{\circ }$, i.e., a right angle. The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.

As a point of clarification, note that C and D are determined by A, B, and the desired ratio; i.e. A and B are not arbitrary points lying on an extension of the diameter of an existing circle.

Calculating the parameters of a circle

Early science, particularly geometry and astronomy/astrology, was connected to the divine for most medieval scholars. The compass in this 13th Century manuscript is a symbol of God's act of Creation, as many believed that there was something intrinsically "divine" or "perfect" that could be found in circles

Given three non-collinear points lying on the circle

$\mathrm {P_{1}} ={\begin{bmatrix}x_{1}\\y_{1}\\z_{1}\end{bmatrix}},\mathrm {P_{2}} ={\begin{bmatrix}x_{2}\\y_{2}\\z_{2}\end{bmatrix}},\mathrm {P_{3}} ={\begin{bmatrix}x_{3}\\y_{3}\\z_{3}\end{bmatrix}}$

Radius

The radius of the circle is given by

$\mathrm {r} ={\frac {\left|P_{1}-P_{2}\right|\left|P_{2}-P_{3}\right|\left|P_{3}-P_{1}\right|}{2\left|\left(P_{1}-P_{2}\right)\times \left(P_{2}-P_{3}\right)\right|}}$

Center

The center of the circle is given by

$\mathrm {P_{c}} =\alpha \,P_{1}+\beta \,P_{2}+\gamma \,P_{3}$

where

$\alpha ={\frac {\left|P_{2}-P_{3}\right|^{2}\left(P_{1}-P_{2}\right)\cdot \left(P_{1}-P_{3}\right)}{2\left|\left(P_{1}-P_{2}\right)\times \left(P_{2}-P_{3}\right)\right|^{2}}}$

$\beta ={\frac {\left|P_{1}-P_{3}\right|^{2}\left(P_{2}-P_{1}\right)\cdot \left(P_{2}-P_{3}\right)}{2\left|\left(P_{1}-P_{2}\right)\times \left(P_{2}-P_{3}\right)\right|^{2}}}$

$\gamma ={\frac {\left|P_{1}-P_{2}\right|^{2}\left(P_{3}-P_{1}\right)\cdot \left(P_{3}-P_{2}\right)}{2\left|\left(P_{1}-P_{2}\right)\times \left(P_{2}-P_{3}\right)\right|^{2}}}$

Plane unit normal

A unit normal of the plane containing the circle is given by

${\hat {n}}={\frac {\left(P_{2}-P_{1}\right)\times \left(P_{3}-P_{1}\right)}{\left|\left(P_{2}-P_{1}\right)\times \left(P_{3}-P_{1}\right)\right|}}$

Parametric Equation

Given the radius, $\mathrm {r}$ , center, $\mathrm {P_{c}}$, a point on the circle, $\mathrm {P_{0}}$ and a unit normal of the plane containing the circle, ${\hat {n}}$, the parametric equation of the circle starting from the point $\mathrm {P_{0}}$ and proceeding counterclockwise is given by the following equation:

$\mathrm {R} \left(s\right)=\mathrm {P_{c}} +\cos \left({\frac {\mathrm {s} }{\mathrm {r} }}\right)\left(P_{0}-P_{c}\right)+\sin \left({\frac {\mathrm {s} }{\mathrm {r} }}\right)\left[{\hat {n}}\times \left(P_{0}-P_{c}\right)\right]$

References

See also

• Area enclosed by a circle
• Ball
• Circular sector
• Degree (angle)
• Descartes' theorem
• Directional statistics
• Isoperimetric theorem
• List of circle topics
• Pi
• Sphere
• Unit circle
• von Mises distribution
• Solar symbol
• Circumscribed circle

External links

• Interactive Standard Form Equation of Circle Click and drag points to see standard form equation in action
• Clifford's Circle Chain Theorems. Step by step presentation of the first theorem. Clifford discovered, in the ordinary Euclidean plane, a "sequence or chain of theorems" of increasing complexity, each building on the last in a natural progression by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
• Munching on Circles at cut-the-knot
• Ron Blond homepage - interactive applets
• Finding the center of a circle with ruler and straightedge Interactive step-by-step animation
• Constructing a circle through 3 given points with ruler and straightedge Interactive step-by-step animation
• calculate circumference and area with your own values
• On-line encyclopedia Springer-Verlag [1]
• Definition and properties of a circle With interactive animation
• Area enclosed by a circle (with interactive animation)