A
chord of a
circleA circle is a simple shape of Euclidean geometry consisting of those 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....
is a
geometricGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
line segmentIn geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
whose endpoints both lie on the
circumferenceThe circumference is the distance around a closed curve. Circumference is a special perimeter.Circumference of a circle:The circumference of a circle is the length around it....
of the circle.
A
secant or a
secant lineA secant line of a curve is a line that intersects two points on the curve. The word secant comes from the Latin secare, to cut.It can be used to approximate the tangent to a curve, at some point P...
is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse. A chord that passes through the circle's center point is the circle's diameter.
Chords of a circle
Among properties of chords of a
circleA circle is a simple shape of Euclidean geometry consisting of those 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....
are the following:
 Chords are equidistant from the center only if their lengths are equal.
 A chord's perpendicular bisector passes through the centre.
 If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).
The area that a circular chord "cuts off" is called a
circular segmentIn geometry, a circular segment is an area of a circle informally defined as an area which is "cut off" from the rest of the circle by a secant or a chord. The circle segment constitutes the part between the secant and an arc, excluding the circle's center...
.
Chords of an ellipse
The midpoints of a set of parallel chords of an ellipse are
collinearA set of points is collinear if they lie on a single line. Related concepts include:In mathematics:...
.
Chords in trigonometry
Chords were used extensively in the early development of
trigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
. The first known trigonometric table, compiled by
HipparchusHipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...
, tabulated the value of the chord function for every 7.5
degreeA degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
s.
PtolemyClaudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
of Alexandria compiled a more extensive table of chords in
his book on astronomyThe Almagest is a 2ndcentury mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
, giving the value of the chord for angles ranging from 1/2 degree to 180 degrees by increments of half a degree.
The chord function is defined geometrically as in the picture to the left. The chord of an
angleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
is the
lengthIn geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
of the chord between two points on a unit circle separated by that angle. The chord function can be related to the modern
sineIn mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
function, by taking one of the points to be (1,0), and the other point to be (cos
, sin
), and then using the
Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
to calculate the chord length:

The last step uses the halfangle formula. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve volume work on chords, all now lost, so presumably a great deal was known about them. The chord function satisfies many identities analogous to wellknown modern ones:
Name  Sinebased  Chordbased 

Pythagorean 


Halfangle 


The halfangle identity greatly expedites the creation of chord tables. Ancient chord tables typically used a large value for the
radiusIn classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
of the circle, and reported the chords for this circle. It was then a simple matter of scaling to determine the necessary chord for any circle. According to G. J. Toomer, Hipparchus used a circle of radius 3438' (= 3438/60 = 57.3). This value is extremely close to
(= 57.29577951...). One advantage of this choice of radius was that he could very accurately approximate the chord of a small angle as the angle itself. In modern terms, it allowed a simple
linear approximationIn mathematics, a linear approximation is an approximation of a general function using a linear function . They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.Definition:Given a twice continuously...
:

Calculating circular chords
The chord of a circle can be calculated using other information:
External links