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Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two sexagesimal (base sixty) digits after the integer part. [2] The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle.
A circular segment (in green) is enclosed between a secant/chord (the dashed line) and the arc whose endpoints equal the chord's (the arc shown above the green area). In geometry , a circular segment or disk segment (symbol: ⌓ ) is a region of a disk [ 1 ] which is "cut off" from the rest of the disk by a straight line.
The value of the two products in the chord theorem depends only on the distance of the intersection point S from the circle's center and is called the absolute value of the power of S; more precisely, it can be stated that: | | | | = | | | | =, where r is the radius of the circle, and d is the distance between the center of the circle and the ...
For modern 64-bit floating-point numbers, the spherical law of cosines formula, given above, does not have serious rounding errors for distances larger than a few meters on the surface of the Earth. [3] The haversine formula is numerically better-conditioned for small distances by using the chord-length relation: [4]
A standard chord length is used: in the UK this is conventionally 30 metres, or sometimes 20 metres. Half chords, i.e. 15 metre or 10 metre intervals, are marked on the datum rail using chalk. The string, which is one full chord long, is then held taut with one end on two marks at each end of a chord, and the offset at the half chord mark measured.
When the sagitta is small in comparison to the radius, it may be approximated by the formula [2] s ≈ l 2 8 r . {\displaystyle s\approx {\frac {l^{2}}{8r}}.} Alternatively, if the sagitta is small and the sagitta, radius, and chord length are known, they may be used to estimate the arc length by the formula
The constant chord theorem is a statement in elementary geometry about a property of certain chords in two intersecting circles. The circles k 1 {\displaystyle k_{1}} and k 2 {\displaystyle k_{2}} intersect in the points P {\displaystyle P} and Q {\displaystyle Q} .
When the arc reaches 60°, the chord length is exactly equal to the number of degrees in the arc, i.e. chord 60° = 60. For arcs of more than 60°, the chord is less than the arc, until an arc of 180° is reached, when the chord is only 120. The fractional parts of chord lengths were expressed in sexagesimal (base 60) numerals. For example ...