There are times in life when the knowledge gained during schooling is very useful. Although during my studies, this information seemed boring and unnecessary. For example, how can you use information about how the length of a chord is found? It can be assumed that for speci alties not related to the exact sciences, such knowledge is of little use. However, there are many examples (from designing a New Year's costume to the complex construction of an airplane) when skills in solving problems in geometry are useful.
The concept of "chord"
This word means "string" in translation from the language of Homer's homeland. It was introduced by mathematicians of the ancient period.
Chord in the section of elementary geometry is a part of a straight line that unites any two points of any curve (circle, parabola or ellipse). In other words, this connecting geometric element is located on a straight line that intersects the given curve at several points. In the case of a circle, the chord length is enclosed between two points of this figure.
Part of a plane bounded by a straight line intersecting a circle and its arc is called a segment. You can note,that as you approach the center, the length of the chord increases. The part of a circle between two points of intersection 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 rest against the points of intersection of the chord with the circle.
Properties and formulas
The chord length of a circle can be calculated from 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 – circle diameter;
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 below:
- Any chords that are the same distance from the center have equal lengths, and the converse is also true.
- All angles that are inscribed in a circle and based on a common segment that connects two points (while their vertices are on the same side of this element) are identical in size.
- The largest chord is the diameter.
- The sum of any two angles, if they are based on a given segment, but their vertices lie on different sides relative to it, is 180o.
- A large chord - compared to a similar but smaller element - lies closer to the middle of this geometric figure.
- All angles that are inscribed and based on the diameter are 90˚.
Other calculations
To find the length of the arc of a circle that lies between the ends of a chord, you can use the Huygens formula. To do this, you need to carry out the following actions:
- Denote the desired value p, and the chord bounding this part of the circle will be called AB.
- Find the midpoint of segment AB and put a perpendicular to it. 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, is in contact with the circle, we denote M.
- Then the segments AM and VM can be called respectively as l and L.
- Arc length can be calculated using the following formula: р≈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%.
Chord length can be used in various fields. For example, when calculating and designing flange connections, which are widely used in technology. You can also see the calculation of this value in ballistics to determine the distance of a bullet and so on.
