Circle

The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the circumference C is related to the radius r and diameter d by:

$${\displaystyle C=2\pi r=\pi d.}$$

Area enclosed by a circle = π × area of the shaded square

As proved by Archimedes, in his Measurement of a Circle, 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,[8] which comes to π multiplied by the radius squared:

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

Equivalently, denoting diameter by d,

$${\displaystyle \mathrm {Area} ={\frac {\pi d^{2}}{4}}\approx 0.7854d^{2},}$$

that is, approximately 79% of the circumscribing square (whose side is of length d).

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.

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

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

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

$${\displaystyle x^{2}+y^{2}=r^{2}.}$$

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

$${\displaystyle {\begin{aligned}x&=a+r\,\cos t,\\y&=b+r\,\sin t,\end{aligned}}}$$

where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the positive x axis.

An alternative parametrisation of the circle is

$${\displaystyle {\begin{aligned}x&=a+r{\frac {1-t^{2}}{1+t^{2}}},\\y&=b+r{\frac {2t}{1+t^{2}}}.\end{aligned}}}$$

In this parameterisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the x axis (see Tangent half-angle substitution). However, this parameterisation works only if t is made to range not only through all reals but also to a point at infinity; otherwise, the leftmost point of the circle would be omitted.

The equation of the circle determined by three points ${\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})}$ not on a line is obtained by a conversion of the 3-point form of a circle equation:

$${\displaystyle {\frac {({\color {green}x}-x_{1})({\color {green}x}-x_{2})+({\color {red}y}-y_{1})({\color {red}y}-y_{2})}{({\color {red}y}-y_{1})({\color {green}x}-x_{2})-({\color {red}y}-y_{2})({\color {green}x}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.}$$

In homogeneous coordinates, each conic section with the equation of a circle has the form

$${\displaystyle x^{2}+y^{2}-2axz-2byz+cz^{2}=0.}$$

It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are called the circular points at infinity.

In polar coordinates, the equation of a circle is

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

where a is the radius of the circle, ${\displaystyle (r,\theta )}$ are the polar coordinates of a generic point on the circle, and ${\displaystyle (r_{0},\phi )}$ are the polar coordinates 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 on the origin, i.e. r₀ = 0, this reduces to r = a. When r₀ = a, or when the origin lies on the circle, the equation becomes

$${\displaystyle r=2a\cos(\theta -\phi ).}$$

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

$${\displaystyle r=r_{0}\cos(\theta -\phi )\pm {\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\theta -\phi )}}.}$$

Note that without the ± sign, the equation would in some cases describe only half a circle.

In the complex plane, a circle with a centre at c and radius r has the equation

$${\displaystyle |z-c|=r.}$$

In parametric form, this can be written as

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

The slightly generalised equation

$${\displaystyle pz{\overline {z}}+gz+{\overline {gz}}=q}$$

for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with ${\displaystyle p=1,\ g=-{\overline {c}},\ q=r^{2}-|c|^{2}}$, since ${\displaystyle |z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}}$. Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.

The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x₁, y₁) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x₁, y₁), so it has the form (x₁ − a)x + (y₁ – b)y = c. Evaluating at (x₁, y₁) determines the value of c, and the result is that the equation of the tangent is

$${\displaystyle (x_{1}-a)x+(y_{1}-b)y=(x_{1}-a)x_{1}+(y_{1}-b)y_{1},}$$

or

$${\displaystyle (x_{1}-a)(x-a)+(y_{1}-b)(y-b)=r^{2}.}$$

If y₁ ≠ b, then the slope of this line is

$${\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}-a}{y_{1}-b}}.}$$

This can also be found using implicit differentiation.

When the centre of the circle is at the origin, then the equation of the tangent line becomes

$${\displaystyle x_{1}x+y_{1}y=r^{2},}$$

and its slope is

$${\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}}{y_{1}}}.}$$

• 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 are:
  • A perpendicular line from the centre of a circle bisects the chord.
  • The line 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 supplementary.
  • 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.
  • Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB.
• 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.[9]
• 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 perpendicular chords intersecting at the same point and is given by 8r² − 4p², where r is the circle radius, and p is the distance from the centre point to the point of intersection.[10]
• The distance 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.[11]^: p.71

• A line drawn perpendicular to a radius through the end point of the radius lying on the circle 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).

Secant–secant theorem

• The chord theorem states that if two chords, CD and EB, intersect at A, then AC × AD = AB × AE.
• If two secants, AE and AD, also cut the circle at B and C respectively, then AC × AD = AB × AE (corollary of the chord theorem).
• A tangent can be considered a limiting case of a secant whose ends are coincident. If a tangent from an external point A meets the circle at F and a secant from the external point A meets the circle at C and D respectively, then AF² = AC × AD (tangent–secant theorem).
• The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side of the chord (tangent chord angle).
• If the angle subtended by the chord at the centre is 90°, then ℓ = r √2, 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 (${\displaystyle {\overset {\frown }{DE}}}$ and ${\displaystyle {\overset {\frown }{BC}}}$). That is, ${\displaystyle 2\angle {CAB}=\angle {DOE}-\angle {BOC}}$, where O is the centre of the circle (secant–secant theorem).

Inscribed-angle theorem

An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180°).

The sagitta is the vertical segment.

The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle.

Given the length y of a chord and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines:

$${\displaystyle r={\frac {y^{2}}{8x}}+{\frac {x}{2}}.}$$

Another proof of this result, which relies only on two chord properties given above, is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know that it is a part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is (2r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2r − x)x = (y / 2)². Solving for r, we find the required result.

• Construct the midpoint M of the diameter.
• Construct the circle with centre M passing through one of the endpoints of the diameter (it will also pass through the other endpoint).

Construct a circle through points A, B and C by finding the perpendicular bisectors (red) of the sides of the triangle (blue). Only two of the three bisectors are needed to find the centre.

• Name the points P, Q and R,
• Construct the perpendicular bisector of the segment PQ.
• Construct the perpendicular bisector of the segment PR.
• Label the point of intersection of these two perpendicular bisectors M. (They meet because the points are not collinear).
• Construct the circle with centre M passing through one of the points P, Q or R (it will also pass through the other two points).

A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If A, B, and C are as above, then the circle of Apollonius for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one:

$${\displaystyle {\bigl |}[A,B;C,P]{\bigr |}=1.}$$

Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio [A, B; C, P] is on the unit circle in the complex plane.

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition

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

is not a circle, but rather a line.

Thus, if A, B, and C are given distinct points in the plane, then the locus of points P satisfying the above equation is called a "generalised circle." It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius.

Illustrations of unit circles (see also superellipse) in different p-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding p).

Defining a circle as the set of points with a fixed distance from a point, different shapes can be considered circles under different definitions of distance. In p-norm, distance is determined by

$${\displaystyle \left\|x\right\|_{p}=\left(\left|x_{1}\right|^{p}+\left|x_{2}\right|^{p}+\dotsb +\left|x_{n}\right|^{p}\right)^{1/p}.}$$

In Euclidean geometry, p = 2, giving the familiar

$${\displaystyle \left\|x\right\|_{2}={\sqrt {\left|x_{1}\right|^{2}+\left|x_{2}\right|^{2}+\dotsb +\left|x_{n}\right|^{2}}}.}$$

In taxicab geometry, p = 1. Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes. While each side would have length ${\displaystyle {\sqrt {2}}r}$ using a Euclidean metric, where r is the circle's radius, its length in taxicab geometry is 2r. Thus, a circle's circumference is 8r. Thus, the value of a geometric analog to ${\displaystyle \pi }$ is 4 in this geometry. The formula for the unit circle in taxicab geometry is ${\displaystyle |x|+|y|=1}$ in Cartesian coordinates and

$${\displaystyle r={\frac {1}{\left|\sin \theta \right|+\left|\cos \theta \right|}}}$$

in polar coordinates.

A circle of radius 1 (using this distance) is the von Neumann neighborhood of its centre.

A circle of radius r for the Chebyshev distance (L_∞ metric) on a plane is also a square with side length 2r parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between L₁ and L_∞ metrics does not generalize to higher dimensions.

In topology, a circle is not limited to the geometric concept, but to all of its homeomorphisms. Two topological circles are equivalent if one can be transformed into the other via a deformation of R³ upon itself (known as an ambient isotopy).[21]