Dimensions of Circle Structural Drawing

Simple bend of Euclidean geometry

This commodity is about the shape and mathematical concept. For other uses, meet Circle (disambiguation).

Circumvolve
A circle (black), which is measured by its circumference (C), diameter (D) in blueish, and radius (R) in blood-red; its centre (O) is in greenish.
Type	Conic section
Symmetry group	O(ii)
Area	πRtwo
Perimeter	C = 2πR

A circumvolve is a shape consisting of all points in a plane that are at a given distance from a given bespeak, the eye; equivalently it is the curve traced out by a point that moves in a aeroplane and so that its distance from a given bespeak is constant. The distance betwixt any bespeak of the circle and the middle is called the radius. Usually, the radius is required to be a positive number. A circle with ${\displaystyle r=0}$ is a degenerate example. This article is about circles in Euclidean geometry, and, in particular, the Euclidean aeroplane, except where otherwise noted.

Specifically, a circle is a unproblematic closed curve that divides the plane into two regions: an interior and an exterior. In everyday utilize, the term "circle" may be used interchangeably to refer to either the purlieus of the figure, or to the whole figure including its interior; in strict technical usage, the circumvolve is just the boundary and the whole figure is called a disc.

A circle may also be defined as a special kind of ellipse in which the 2 foci are ancillary, the eccentricity is 0, and the semi-major and semi-minor axes are equal; or the 2-dimensional shape enclosing the nigh area per unit perimeter squared, using calculus of variations.

Euclid's definition

A circumvolve is a plane figure divisional by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is chosen its circumference and the signal, its eye.

Topological definition

In the field of topology, a circle isn't express to the geometric concept, but to all of its homeomorphisms. Two topological circles are equivalent if 1 can be transformed into the other via a deformation of R ⁱⁱⁱ upon itself (known as an ambience isotopy).^[ii]

Terminology

• Annulus: a ring-shaped object, the region bounded by two concentric circles.
• Arc: any continued part of a circle. Specifying 2 end points of an arc and a center allows for ii arcs that together make up a full circle.
• Centre: the point equidistant from all points on the circle.
• Chord: a line segment whose endpoints lie on the circle, thus dividing a circumvolve into two segments.
• Circumference: the length of 1 circuit along the circle, or the distance around the circle.
• Diameter: a line segment whose endpoints prevarication on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.
• Disc: the region of the airplane bounded past a circle.
• Lens: the region common to (the intersection of) two overlapping discs.
• Passant: a coplanar straight line that has no point in common with the circle.
• Radius: a line segment joining the centre of a circumvolve with any single indicate on the circle itself; or the length of such a segment, which is half (the length of) a diameter.
• Sector: a region bounded past 2 radii of equal length with a mutual center and either of the 2 possible arcs, determined by this center and the endpoints of the radii.
• Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used merely for regions not containing the center of the circumvolve to which their arc belongs to.
• Secant: an extended chord, a coplanar directly line, intersecting a circle in ii points.
• Semicircle: one of the 2 possible arcs determined by the endpoints of a diameter, taking its midpoint every bit center. In not-technical common usage it may mean the interior of the two dimensional region bounded by a diameter and ane of its arcs, that is technically called a one-half-disc. A half-disc is a special case of a segment, namely the largest one.
• Tangent: a coplanar direct line that has one unmarried point in common with a circumvolve ("touches the circumvolve at this point").

All of the specified regions may exist considered as open up, that is, not containing their boundaries, or as airtight, including their respective boundaries.

Chord, secant, tangent, radius, and diameter

History

The compass in this 13th-century manuscript is a symbol of God's human action of Creation. Observe also the circular shape of the halo.

The discussion circle derives from the Greek κίρκος/κύκλος (kirkos/kuklos), itself a metathesis of the Homeric Greek κρίκος (krikos), pregnant "hoop" or "band".^[iii] The origins of the words circus and circuit are closely related.

Circular piece of silk with Mongol images

The circle has been known since before the kickoff of recorded history. Natural circles would accept been observed, such as the Moon, Sun, and a short plant stalk bravado in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of mod machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus.

Early science, peculiarly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be institute in circles.^{[4] [5]}

Some highlights in the history of the circumvolve are:

• 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256 / 81 (iii.16049...) every bit an approximate value of π.^[vi]

• 300 BCE – Book 3 of Euclid's Elements deals with the properties of circles.
• In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how information technology is different from any drawing, words, definition or explanation.
• 1880 CE – Lindemann proves that π is transcendental, effectively settling the millennia-one-time problem of squaring the circle.^[7]

Analytic results

Circumference

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 bore d by:

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

Area enclosed

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

As proved past 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'southward circumference and whose acme equals the circle's radius,^[8] which comes to π multiplied past the radius squared:

${\displaystyle \mathrm {Surface area} =\pi r^{2}.\,}$

Equivalently, denoting diameter by d,

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

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.

Equations

Cartesian coordinates

Circumvolve of radius r = ane, centre (a,b) = (1.ii, −0.5)

Equation of a circle

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

${\displaystyle (10-a)^{2}+(y-b)^{2}=r^{two}.}$

This equation, known as the equation of the circle, follows from the Pythagorean theorem applied to any point on the circumvolve: every bit shown in the next diagram, the radius is the hypotenuse of a correct-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

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

Parametric course

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

${\displaystyle ten=a+r\,\cos t,}$

${\displaystyle y=b+r\,\sin t,}$

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

An alternative parametrisation of the circle is

${\displaystyle ten=a+r{\frac {1-t^{2}}{1+t^{two}}},}$

${\displaystyle y=b+r{\frac {2t}{ane+t^{2}}}.}$

In this parameterisation, the ratio of t to r tin can exist interpreted geometrically as the stereographic project of the line passing through the center parallel to the x axis (see Tangent half-angle substitution). However, this parameterisation works simply if t is made to range non only through all reals simply besides to a point at infinity; otherwise, the leftmost point of the circle would be omitted.

iii-signal grade

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

${\displaystyle {\frac {({\colour {light-green}ten}-x_{1})({\color {green}x}-x_{2})+({\colour {red}y}-y_{1})({\colour {red}y}-y_{ii})}{({\color {red}y}-y_{1})({\color {green}10}-x_{ii})-({\color {red}y}-y_{2})({\colour {green}10}-x_{ane})}}={\frac {(x_{3}-x_{i})(x_{3}-x_{2})+(y_{three}-y_{1})(y_{3}-y_{ii})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{three}-y_{2})(x_{3}-x_{1})}}.}$

Homogeneous grade

In homogeneous coordinates, each conic department with the equation of a circle has the grade

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

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

Polar coordinates

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 circumvolve, and ${\displaystyle (r_{0},\phi )}$ are the polar coordinates of the centre of the circumvolve (i.e., r ₀ is the distance from the origin to the centre of the circumvolve, 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. r ₀ = 0, this reduces to simply r = a . When r ₀ = a , or when the origin lies on the circumvolve, the equation becomes

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

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

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

Notation that without the ± sign, the equation would in some cases draw merely one-half a circle.

Complex plane

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

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

In parametric grade, 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 thou is sometimes called a generalised circle. This becomes the higher up equation for a circle with ${\displaystyle p=1,\ chiliad=-{\overline {c}},\ q=r^{two}-|c|^{2}}$ , since ${\displaystyle |z-c|^{two}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}}$ . Not all generalised circles are really circles: a generalised circle is either a (true) circle or a line.

Tangent lines

The tangent line through a indicate P on the circumvolve is perpendicular to the diameter passing through P. If P = (x ₁, y ₁) and the circumvolve has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x ₁, y ₁), so it has the form (x ₁ − a)x + (y ₁ – b)y = c . Evaluating at (x _i, y ₁) determines the value of c, and the result is that the equation of the tangent is

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

or

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

If y ₁ ≠ b , and then the gradient of this line is

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

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

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

and its slope is

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

Properties

• The circle is the shape with the largest area for a given length of perimeter (run across Isoperimetric inequality).
• The circle is a highly symmetric shape: every line through the center 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 solitary is the circle group T.
• All circles are similar.
  • A circle circumference and radius are proportional.
  • The expanse enclosed and the square of its radius are proportional.
  • The constants of proportionality are 2π and π respectively.
• The circumvolve that is centred at the origin with radius 1 is called the unit of measurement circumvolve.
  • Idea of as a great circle of the unit sphere, information technology becomes the Riemannian circumvolve.
• Through whatever three points, non all on the aforementioned line, there lies a unique circle. In Cartesian coordinates, information technology 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 eye of a circle if and just if they are equal in length.
• The perpendicular bisector of a chord passes through the heart 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 heart bisecting a chord is perpendicular to the chord.
• If a central angle and an inscribed bending of a circle are subtended by the same chord and on the aforementioned side of the chord, then the central angle is twice the inscribed bending.
• 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, and then they are supplementary.
  • For a cyclic quadrilateral, the outside bending is equal to the interior opposite angle.
• An inscribed angle subtended by a diameter is a correct angle (encounter 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 1 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, and then ab = cd .
• If the intersection of whatsoever two perpendicular chords divides 1 chord into lengths a and b and divides the other chord into lengths c and d, then a ² + b ² + c ² + d ² equals the foursquare of the diameter.^[ix]
• 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 signal and is given past 8r ^two − 4p ², where r is the circle radius, and p is the distance from the centre signal to the point of intersection.^[10]
• The distance from a point on the circle to a given chord times the diameter of the circumvolve equals the production of the distances from the betoken to the ends of the chord.^{[eleven] : p.71}

Tangent

• A line fatigued perpendicular to a radius through the stop betoken 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 circumvolve passes through the centre of the circumvolve.
• 2 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 outside bespeak P, then denoting the centre as O, the angles ∠BOA and ∠BPA are supplementary.
• If Advertising is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = 1 / 2 arc(AQ).

Theorems

• The chord theorem states that if two chords, CD and EB, intersect at A, then Air-conditioning × AD = AB × AE .
• If ii secants, AE and AD, too cutting the circle at B and C respectively, so AC × Advertisement = AB × AE (corollary of the chord theorem).
• A tangent can be considered a limiting instance of a secant whose ends are ancillary. If a tangent from an external point A meets the circle at F and a secant from the external point A meets the circumvolve at C and D respectively, then AF ² = AC × Advert (tangent–secant theorem).
• The angle betwixt a chord and the tangent at one of its endpoints is equal to i half the angle subtended at the heart of the circle, on the opposite side of the chord (tangent chord angle).
• If the angle subtended by the chord at the heart is 90°, then ℓ = r √2 , where ℓ is the length of the chord, and r is the radius of the circumvolve.
• If 2 secants are inscribed in the circumvolve as shown at correct, then the measurement of angle A is equal to i half the difference of the measurements of the enclosed arcs ( ${\displaystyle {\overset {\pout }{DE}}}$ and ${\displaystyle {\overset {\frown }{BC}}}$ ). That is, ${\displaystyle 2\angle {CAB}=\bending {DOE}-\bending {BOC}}$ , where O is the centre of the circle (secant–secant theorem).

Inscribed angles

An inscribed angle (examples are the blueish and green angles in the effigy) is exactly half the corresponding fundamental bending (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 bore is a right angle (since the central angle is 180°).

Sagitta

The sagitta is the vertical segment.

The sagitta (also known every bit the versine) is a line segment fatigued 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 tin be used to calculate the radius of the unique circle that volition fit around the ii 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 10, since the sagitta intersects the midpoint of the chord, we know that information technology is a office of a diameter of the circumvolve. Since the diameter is twice the radius, the "missing" part of the diameter is (2r − x ) in length. Using the fact that one function of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (twor − x)x = (y / 2)² . Solving for r, we observe the required upshot.

Compass and straightedge constructions

There are many compass-and-straightedge constructions resulting in circles.

The simplest and about basic is the construction given the centre of the circumvolve and a point on the circle. Place the fixed leg of the compass on the centre point, the movable leg on the point on the circle and rotate the compass.

Construction with given diameter

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

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

Construction through three noncollinear points

• 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 indicate of intersection of these two perpendicular bisectors M . (They meet considering the points are not collinear).
• Construct the circle with centre M passing through one of the points P , Q or R (it volition also laissez passer through the other ii points).

Circumvolve of Apollonius

Apollonius' definition of a circle: d ₁/d ₂ constant

Apollonius of Perga showed that a circumvolve may also be divers as the set of points in a plane having a abiding ratio (other than i) of distances to two fixed foci, A and B.^{[12] [xiii]} (The set of points where the distances are equal is the perpendicular bisector of segment AB, a line.) That circumvolve is sometimes said to be drawn nearly two points.

The proof is in 2 parts. First, one must prove that, given ii foci A and B and a ratio of distances, any point P satisfying the ratio of distances must autumn on a item circle. Allow C be another point, also satisfying the ratio and lying on segment AB. By the bending bisector theorem the line segment PC will bifurcate the interior angle APB, since the segments are similar:

${\displaystyle {\frac {AP}{BP}}={\frac {Air-conditioning}{BC}}.}$

Analogously, a line segment PD through some point D on AB extended bisects the corresponding exterior angle BPQ where Q is on AP extended. Since the interior and exterior angles sum to 180 degrees, the angle CPD is exactly 90 degrees; that is, a right angle. The set of points P such that angle CPD is a right angle forms a circle, of which CD is a bore.

Second, run across^{[14] : p.15} for a proof that every point on the indicated circle satisfies the given ratio.

Cantankerous-ratios

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

${\displaystyle {\big |}[A,B;C,P]{\big |}=ane.}$

Stated another way, P is a indicate on the circumvolve of Apollonius if and simply if the cantankerous-ratio [A, B; C, P] is on the unit of measurement circumvolve in the complex plane.

Generalised circles

If C is the midpoint of the segment AB, then the drove of points P satisfying the Apollonius status

${\displaystyle {\frac {|AP|}{|BP|}}={\frac {|Air-conditioning|}{|BC|}}}$

is not a circumvolve, 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 in a higher place equation is called a "generalised circumvolve." Information technology may either be a true circle or a line. In this sense a line is a generalised circle of space radius.

Inscription in or circumscription almost other figures

In every triangle a unique circumvolve, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle.^[xv]

About every triangle a unique circle, called the circumcircle, can be confining such that it goes through each of the triangle's three vertices.^[16]

A tangential polygon, such as a tangential quadrilateral, is whatsoever convex polygon within which a circumvolve can be inscribed that is tangent to each side of the polygon.^[17] Every regular polygon and every triangle is a tangential polygon.

A cyclic polygon is any convex polygon about which a circle can be circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a bicentric polygon.

A hypocycloid is a curve that is inscribed in a given circle past tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.

Limiting instance of other figures

The circle can exist viewed as a limiting case of each of various other figures:

• A Cartesian oval is a ready of points such that a weighted sum of the distances from whatsoever of its points to 2 stock-still points (foci) is a constant. An ellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circumvolve. A circle is likewise a unlike special case of a Cartesian oval in which ane of the weights is zero.
• A superellipse has an equation of the form ${\displaystyle \left|{\frac {x}{a}}\right|^{n}\!+\left|{\frac {y}{b}}\right|^{n}\!=i}$ for positive a, b, and n. A supercircle has b = a . A circle is the special case of a supercircle in which n = 2.
• A Cassini oval is a set of points such that the production of the distances from any of its points to ii fixed points is a constant. When the 2 fixed points coincide, a circumvolve results.
• A bend of abiding width is a figure whose width, defined as the perpendicular distance between two singled-out parallel lines each intersecting its boundary in a unmarried bespeak, is the same regardless of the direction of those two parallel lines. The circle is the simplest instance of this blazon of figure.

In other p-norms

Illustrations of unit of measurement circles (run into also superellipse) in different p -norms (every vector from the origin to the unit circumvolve has a length of 1, the length existence calculated with length-formula of the corresponding p ).

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

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

In Euclidean geometry, p = 2, giving the familiar

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

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'due south radius, its length in taxicab geometry is 2r. Thus, a circle's circumference is viiir. 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

${\displaystyle r={\frac {one}{|\sin \theta |+|\cos \theta |}}}$

in polar coordinates.

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

A circumvolve of radius r for the Chebyshev distance (Fifty_∞ metric) on a plane is also a square with side length 2r parallel to the coordinate axes, and so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence betwixt L_one and 50_∞ metrics does not generalize to higher dimensions.

Locus of constant sum

Consider a finite ready of ${\displaystyle n}$ points in the plane. The locus of points such that the sum of the squares of the distances to the given points is constant is a circumvolve, whose eye is at the centroid of the given points.^[eighteen] A generalization for college powers of distances is obtained if under ${\displaystyle north}$ points the vertices of the regular polygon ${\displaystyle P_{n}}$ are taken.^[19] The locus of points such that the sum of the ${\displaystyle (2m)}$ -th ability of distances ${\displaystyle d_{i}}$ to the vertices of a given regular polygon with circumradius ${\displaystyle R}$ is constant is a circumvolve, if

${\displaystyle \sum _{i=1}^{north}d_{i}^{2m}>nR^{2m}}$ , where ${\displaystyle m}$ =1,2,…, ${\displaystyle due north}$ -1;

whose center is the centroid of the ${\displaystyle P_{n}}$ .

In the case of the equilateral triangle, the loci of the constant sums of the second and fourth powers are circles, whereas for the foursquare, the loci are circles for the abiding sums of the second, fourth, and sixth powers. For the regular pentagon the constant sum of the 8th powers of the distances will exist added and so forth.

Squaring the circle

Squaring the circle is the problem, proposed past aboriginal geometers, of amalgam a square with the same surface area every bit a given circumvolve by using merely a finite number of steps with compass and straightedge.

In 1882, the task was proven to be impossible, as a upshot of the Lindemann–Weierstrass theorem, which proves that pi (π) is a transcendental number, rather than an algebraic irrational number; that is, it is not the root of whatever polynomial with rational coefficients. Despite the impossibility, this topic continues to be of interest for pseudomath enthusiasts.

Significance in art and symbolism

From the fourth dimension of the earliest known civilisations – such as the Assyrians and ancient Egyptians, those in the Indus Valley and along the Yellow River in China, and the Western civilisations of ancient Hellenic republic and Rome during classical Antiquity – the circle has been used directly or indirectly in visual art to convey the artist's message and to limited sure ideas. However, differences in worldview (beliefs and civilisation) had a great impact on artists' perceptions. While some emphasised the circumvolve'south perimeter to demonstrate their democratic manifestation, others focused on its centre to symbolise the concept of cosmic unity. In mystical doctrines, the circle mainly symbolises the infinite and cyclical nature of beingness, but in religious traditions it represents heavenly bodies and divine spirits. The circumvolve signifies many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, remainder, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the employ of symbols, for example, a compass, a halo, the vesica piscis and its derivatives (fish, eye, aureole, mandorla, etc.), the ouroboros, the Dharma wheel, a rainbow, mandalas, rose windows and so forth.^[20]

Come across also

References

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3. ^ krikos Archived 2013-xi-06 at the Wayback Machine, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
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8. ^ Katz, Victor J. (1998), A History of Mathematics / An Introduction (second ed.), Addison Wesley Longman, p. 108, ISBN978-0-321-01618-8
9. ^ Posamentier and Salkind, Challenging Problems in Geometry, Dover, second edition, 1996: pp. 104–105, #4–23.
10. ^ College Mathematics Periodical 29(4), September 1998, p. 331, trouble 635.
11. ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007.
12. ^ Harkness, James (1898). "Introduction to the theory of analytic functions". Nature. 59 (1530): 30. Bibcode:1899Natur..59..386B. doi:ten.1038/059386a0. S2CID 4030420. Archived from the original on 2008-10-07.
13. ^ Ogilvy, C. Stanley, Excursions in Geometry, Dover, 1969, 14–17.
14. ^ Altshiller-Court, Nathan, Higher Geometry, Dover, 2007 (orig. 1952).
15. ^ Incircle – from Wolfram MathWorld Archived 2012-01-21 at the Wayback Machine. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
16. ^ Circumcircle – from Wolfram MathWorld Archived 2012-01-20 at the Wayback Machine. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
17. ^ Tangential Polygon – from Wolfram MathWorld Archived 2013-09-03 at the Wayback Car. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
18. ^ Apostol, Tom; Mnatsakanian, Mamikon (2003). "Sums of squares of distances in m-infinite". American Mathematical Monthly. 110 (six): 516–526. doi:ten.1080/00029890.2003.11919989. S2CID 12641658.
19. ^ Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv:2010.12340.
20. ^ Abdullahi, Yahya (Oct 29, 2019). "The Circle from East to West". In Charnier, Jean-François (ed.). The Louvre Abu Dhabi: A World Vision of Art. Rizzoli International Publications, Incorporated. ISBN9782370741004.

Further reading

• Pedoe, Dan (1988). Geometry: a comprehensive course . Dover. ISBN9780486658124.
• "Circle" in The MacTutor History of Mathematics archive

External links

Wikiquote has quotations related to: Circles

• "Circle", Encyclopedia of Mathematics, Ems Press, 2001 [1994]
• Circle at PlanetMath.
• Weisstein, Eric W. "Circle". MathWorld.
• "Interactive Java applets". for the backdrop of and elementary constructions involving circles
• "Interactive Standard Form Equation of Circumvolve". Click and elevate points to see standard form equation in action
• "Munching on Circles". cutting-the-knot

karraftyrand.blogspot.com

Source: https://en.wikipedia.org/wiki/Circle

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