Part of circle
Circle
Circle
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(where r = radius)
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Tycho crater, one of many examples of circles that arise in nature
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.
Contents
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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
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Arc, sector, and segment
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History
The word "circle" derives
from the Greek, kirkos "a circle," from the base ker-
which means to turn or bend. The origins of the words "circus" and "circuit" are
closely related.
The circle has been known since
before the beginning of recorded history. Natural circles would have been
observed, such as the Moon, Sun, and a short plant stalk blowing 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 modern civilisation possible. In
mathematics, the study of the circle has helped inspire the development of
geometry, astronomy, and calculus.
Early science,
particularly 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
found in circles.[citation
needed]
Tughrul Tower from inside
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Circles on an old Arabic
astronomical drawing
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Some highlights in the history of
the circle are:
- 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256 / 81 (3.16049...) as an approximate value of Ï€.[1]
- 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 it is different from any drawing, words, definition or explanation.
- 1880 CE– Lindemann proves that Ï€ is transcendental, effectively settling
the millennia-old problem of squaring
the circle.[2]
Analytic
results
Length
of circumference
Further information: Pi
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:
Area
enclosed
Area enclosed by a circle = Ï€ × area
of the shaded square
Main article: Area of a disk
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:
Equivalently, denoting diameter by d,
that is, approximately 79 percent 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
Circle of radius r = 1,
centre (a, b) = (1.2, −0.5)
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
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
The equation can be written in parametric form using the trigonometric functions sine and cosine as
where t is a parametric variable in the range 0 to 2Ï€, interpreted geometrically as the
angle that the ray from (a, b) to (x, y)
makes with the x-axis. An alternative parametrisation of the circle is:
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
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.
Polar
coordinates
In polar coordinates
the equation of a circle is:
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
In the general case, the equation
can be solved for r, giving
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.
Tangent
lines
Main article: Tangent
lines to circles
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
or
If y1 ≠ b
then the slope of this line is
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
and its slope is
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.[4]
- 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).[5]
- 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.[6]:p.71
Sagitta
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 which will fit around the two lines:
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 it is 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.
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).
Theorems
Secant-secant theorem
See also: Power of a point
- The chord theorem states that if two chords, CD and EB, intersect at A, then CA × DA = EA × BA.
- 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 DC2 = DG × DE. (Tangent-secant theorem.)
- If two secants, DG and DE, also cut the circle at H and F respectively, then DH × DG = DF × DE. (Corollary of the tangent-secant theorem.)
- The angle between a tangent and chord is equal to one half the subtended angle on the opposite side of the chord (Tangent Chord Angle).
- If the angle subtended by the chord at the centre is 90 degrees 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 (DE and BC). This is the secant-secant theorem.
Inscribed
angles
See also: Inscribed angle 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 degrees).
Circle
of Apollonius
Main article: Apollonian circles
Apollonius' definition of a circle: d1
/ d2 constant
Apollonius of Perga showed that a circle may also be defined as the set of
points in a plane having a constant ratio (other than 1) of distances to
two fixed foci, A and B.[7][8]
(The set of points where the distances are equal is the perpendicular bisector
of A and B, a line.) That circle is sometimes said to be drawn about
two points.
The proof is in two parts. First,
one must prove that, given two foci A and B and a ratio of
distances, any point P satisfying the ratio of distances must fall on a
particular circle. Let C be another point, also satisfying the ratio and
lying on segment AB. By the angle bisector theorem the line segment PC will bisect the interior angle
APB, since the segments are similar:
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,
i.e., a right angle. The set of points P such that angle CPD is a
right angle forms a circle, of which CD is a diameter.
Second, see[9]:p.15 for a
proof that every point on the indicated circle satisfies the given ratio.
Cross-ratios
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:
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.
Generalised
circles
See also: Generalised circle
If C is the midpoint of the
segment AB, then the collection of points P satisfying the
Apollonius condition
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.
Circles
inscribed in or circumscribed about other figures
In every triangle a unique
circle, called the incircle, can be inscribed such that it is tangent to each
of the three sides of the triangle.[10]
About every triangle a unique
circle, called the circumcircle, can be circumscribed such that it goes through each of the
triangle's three vertices.[11]
A tangential polygon, such as a tangential
quadrilateral, is any convex polygon
within which a circle can be inscribed that is tangent to each side of the
polygon.[12]
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.
A hypocycloid
is a curve that is inscribed in a given circle by tracing a fixed point on a
smaller circle that rolls within and tangent to the given circle.
Circle
as limiting case of other figures
The circle can be viewed as a
limiting case of each of various other figures:
- A Cartesian oval is a set of points such that a weighted sum of the distances from any of its points to two fixed 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 circle. A circle is also a different special case of a Cartesian oval in which one of the weights is zero.
- A superellipse has an equation of the form 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 product of the distances from any of its points to two fixed points is a constant. When the two fixed points coincide, a circle results.
- A curve of constant width is a figure whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines. The circle is the simplest example of this type of figure.
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