Chord length: basic concepts

There are cases in life when the knowledge gainedduring school education, are very useful. Although during study this information seemed boring and unnecessary. For example, how can you use information about how the chord length is located? It can be assumed that for specialties not related to the exact sciences, such knowledge is of little use. However, you can give many examples (from designing a New Year's costume to a complex airplane device), when the skills in solving problems in geometry are not superfluous.

The concept of "chord"

This word means "string" in the translation from the language of the homeland of Homer. It was introduced by mathematicians of the ancient period.

The chorda is denoted in the elementarygeometry part of a straight line that combines any two points of any curve (circle, parabola or ellipse). In other words, this connecting geometric element is on a straight line that intersects the given curve at several points. In the case of a circle, the length of the chord is enclosed between two points of this figure.

Part of the plane bounded by a straight line,Crossing a circle, and its arc is called a segment. It can be noted that the chord length increases with the approach to the center. A part of a circle between two points of intersection of a given line is called an arc. Its measure of measurement is the central angle. The vertex of this geometric figure is in the middle of the circle, and the sides rest on the intersection points of the chord with the circle.

Properties and formulas

The length of the chord of a circle can be calculated by the following conditional expressions:

L = D × Sinβ or L = D × Sin (1 / 2α), where β is the angle at the vertex of the inscribed triangle;

D is the diameter of the circle;

α is the central angle.

You can select some properties of this segment, as well as other figures associated with it. These points are listed in the following list:

Any chords that are at the same distance from the center have equal lengths, and the converse statement is also true.
All angles that are inscribed in a circle and are supported by a common segment that combines two points (their vertices are on one side of this element) are identical in magnitude.
The largest chord is the diameter.
The sum of any two angles, if they are supported by a given segment, but their vertices lie in different sides with respect to it, is 180^about.
The large chord - in comparison with a similar, but smaller element - lies closer to the middle of this geometric figure.
All angles that are inscribed and supported by a diameter are 90 °.

Other calculations

To find the length of the arc of the circle, which is enclosed between the ends of the chord, you can use the Huygens formula. For this it is necessary to carry out the following actions:

Denote the required value of p, and the chord that bounds this part of the circle will have the name AB.
We find the middle of the segment AB and put it to itperpendicular. It can be noted that the diameter of a circle drawn through the center of the chord forms a right angle with it. The converse is also true. In this case, the point where the diameter, passing through the middle of the chord, touches the circle, is denoted by M.
Then the segments AM and BM can be called respectively, as l and L.
The arc length can be calculated from the followingformula: p≈2l + 1/3 (2l-L). It can be noted that the relative error of this expression increases with increasing angle. Thus, at 60 ° it is 0.5%, and for an arc equal to 45 ° this value decreases to 0.02%.

The length of the chord can be used in variousspheres. For example, in the calculation and design of flange connections, which are widely used in engineering. You can also see the calculation of this value in ballistics to determine the distance of the bullet flight and so on.