There are times in life when the knowledge gained during schooling is very useful. Although during the study, this information seemed boring and unnecessary. For example, how can you use information about how the chord length is? It can be assumed that for specialties not related to the exact sciences, such knowledge is of little use. However, many examples can be cited (from designing a New Year's costume to the complex design of an airplane) when skills in solving geometry problems are not out of place.
The concept of "chord"
This word means "string" in translation from the language of the homeland of Homer. It was introduced by mathematicians of the ancient period.
In the section of elementary geometry, chords denote the 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 located on a line intersecting a given curve at several points. In the case of a
circle, the length of the chord is enclosed between two points of this figure.
The part of the plane bounded by a straight line intersecting a circle and its arc is called a segment. It can be noted that with approaching the center, the length of the chord increases. The part of a circle located between two intersection points of a given line is called an arc. Its measure is the central angle. The top of this geometric figure is in the middle of the circle, and the sides abut at the points of intersection of the chord with the circle.
Properties and formulas
The circumference of a chord can be calculated using the following conditional expressions:
L = D × Sinβ or L = D × Sin (1/2 α), 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 highlight some properties of this segment, as well as other figures associated with it. These points are listed in the following list:
- Any chords located at the same distance from the center have equal lengths, and the converse is also true.
- All angles that are inscribed in a circle and rely on a common segment that combines two points (while their vertices are on the same side of this element) are identical in magnitude.
- The largest chord is the diameter.
- The sum of any two angles, if they rely on a given segment, but at the same time their vertices lie on different sides relative to it, is 180 °.
- A large chord - compared with a similar, but smaller element - lies closer to the middle of this geometric figure.
- All angles that are inscribed and rely on the diameter are 90˚.
Other calculations
To find the length of the arc of a circle that is enclosed between the ends of the chord, you can use the Huygens formula. For this, it is necessary to carry out such actions:
- We denote the desired value of p, and the chord bounding this part of the circle will be called AB.
- Find the middle of the segment AB and put a perpendicular to it. It can be noted that the diameter of the circle drawn through the center of the chord forms a right angle with it. The converse is also true. Moreover, the point where the diameter, passing through the middle of the chord, is in contact with the circle, we denote M.
- Then the segments AM and VM can be called, respectively, as l and L.
- The length of the arc can be calculated by the following formula: p≈2l + 1/3 (2l-L). It can be noted that the relative error of this expression increases with increasing angle. So, at 60˚ it is 0.5%, and for an arc equal to 45˚, this value decreases to 0.02%.
The chord length can be used in various fields. For example, in the calculations and design of flange joints, which are widely used in engineering. You can also see the calculation of this value in ballistics to determine the distance of a bullet’s flight and so on.