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 fixed point, the centre. The points can only be those that are part of a conic section; within the set of a plane normal to the axis of a right cone. Circles are simple closed curves, dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference(C). Usually, however, the circumference 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. Circles are named by their centre, e.g. Circle O. A circle curves to 360 degrees.

1 Analytic results
2 Properties
3 Inscribed angles
4 An alternative definition of a circle
5 Numbers and the circle
6 See also
7 External links

[edit] Analytic results

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

Approximating the area of a circle by regular polygons

Area of a circle using infinitesimal area element

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

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

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

a x 2 + a y 2 + 2 b 1 x z + 2 b 2 y z + c z 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:

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

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:

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

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.

All circles are similar; as a consequence, a circle's circumference and radius are proportional, as are its area and the square of its radius. The constants of proportionality are 2π and π, respectively.

In other words:

Length of a circle's circumference is:

$c = 2\pi \cdot r$

The area enclosed by a circle is:

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

Diameter of a circle is:

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

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

The formula for the area of a circle can be derived from the formula for the circumference and the formula for the area of a triangle, as follows. Imagine a regular hexagon (six-sided figure) divided into equal triangles, with their apices at the centre of the hexagon. The area of the hexagon may be found by the formula for triangle area by adding up the lengths of all the triangle bases (on the exterior of the hexagon), multiplying by the height of the triangles (distance from the middle of the base to the centre) and dividing by two. This is an approximation of the area of a circle. Then imagine the same exercise with an octagon (eight-sided figure), and the approximation is a little closer to the area of a circle. As a regular polygon with more and more sides is divided into triangles and the area calculated from this, the area becomes closer and closer to the area of a circle. In the limit, the sum of the bases approaches the circumference 2πr, and the triangles' height approaches the radius r. Multiplying the circumference and radius and dividing by 2, we get the area, π r².

The formula for the area of circle can also be derived by using an infinitesimal area element $d A$ and integrating it over the whole circle.

In Chinese mathematics the area of a circle is usually expressed as:

A=c/2•d/2

where A is the area, c is the circumference, and d is the diameter. c=2πr, and d=2r, so c/2•d/2=πr²=A, the familiar identity for area.

[edit] Properties

The circle is the shape with the highest area for a given length of perimeter.
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 alone is the circle group T.

[edit] Chord properties

Chords equidistant from the centre of a circle are equal (length).
Equal (length) chords are equidistant from the center.
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 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 semicircle is a right angle.

[edit] 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

[edit] 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 like so

then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC).

[edit] Inscribed angles

Inscribed angle theorem

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

[edit] An alternative definition of a circle

$\frac{d1}{d2}=\textrm{constant}$ 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

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

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.

[edit] Numbers and the circle

The division of the circle into 360 degrees dates back to ancient India, as found in the Rig Veda:

Twelve spokes, one wheel, navels three.

Who can comprehend this?

On it are placed together

three hundred and sixty like pegs.

They shake not in the least.

(Dirghatama, Rig Veda 1.164.48)

This division is used in mathematics, but also in geography and in astronomy to measure the celestial sphere and equator (both in terms of latitude and longitude).

The enneagram expresses the circle as equal to 9 integers. The structure of the enneagram is based partly on the primary triangle of the circle at 0/360 degrees, 120 degrees and 240 degrees of the circle. In terms of integers, these points of the circle correspond to the numbers 0/9, 3 and 6. The rest of the Enneagram structure consists of connections between the other 6 integers of the 9-based number system - determined by the fraction 1/7 = 0.142857 (repeating).

[edit] See also

Wikimedia Commons has media related to:

Circle

[edit] External links

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]

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Categories: Circles | Geometric shapes | Curves | Conic sections | Pi