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Chord Length Formula

The chord of any circle is an important term. It is defined as the line segment joining any two points on the circumference of the circle, not passing through its centre. Therefore, the diameter is the longest chord of a given circle, as it passes through the centre of the circle.  Calculation of the length of the chord is sometimes very important in mathematics. This article will explain the chord length formula with examples. Let us learn it!

What is a Chord in a Circle?

A chord is the line segment in a circle, which connects any two points on the circumference of the circle. The same two points are connected by the curve in the form of the corresponding arc in the circle.

In the given circle having ‘O’ as the centre, AB represents the diameter of the circle i.e. the longest chord, ‘OE’ will be the radius of the circle and line CD represents a chord of the circle, whereas curve CD will be the arc.

Chord Length Formula

Source: en.wikipedia.org

The Formula for Chord Length

We may determine the length of the chord from the length of the radius and the angle made by the lines connecting the circle’s centre to the two ends of the chord CD.

We may also calculate the chord length if we know both the radius and the length of the right bisector. It is due to the fact that perpendicular drawn from centre O on chord CD will be the bisector of CD. Thus using Pythagoras theorem, we may find the length of the chord CD easily.

Therefore, the two basic formulas for finding the length of the chord of a circle are as follows

  1. Chord Length Using Perpendicular Distance from the Centre of the circle: \(C_{len}= 2 \times \sqrt {(r^{2} –d^{2}}\)
  2. Chord Lenth Using Trigonometry with angle \theta: \(C_{len}=   2 \times r \times sin(\frac{\theta}{2})\)
r It is the radius of the circle.
d It is the perpendicular distance from the chord to the centre
\(C_{len}\) Length of the chord
\(\theta\) It is the angle made at the centre by the chord under consideration

Some Important Facts for the Chord of a Circle:

If we try to find out the relationship between different chords and the angle subtended by them on the centre of the circle, there are some facts are:

  • Chords which are equal in length will subtend equal angles at the centre of the circle.
  • If the angles subtended by chords in a circle are equal in the measurement, then the length of the chords is equal.
  • Equal chords of any circle are at the equidistant from the centre of the circle.

Solved Examples for Chord Length Formula

Q.1: Find out the length of the chord of a circle with radius 7 cm. Also, the perpendicular distance from the chord to the centre is 4 cm. Use chord length formula.

Solution: Here given parameters are as follows:

Radius, r = 7 cm

Perpendicular distance from the centre to the chord, d = 4 cm

Now, using the formula for chord length as given:

\(C_{len}= 2 \times \sqrt {(r^{2} –d^{2}}\\\)

\(C_{len}= 2 \times \sqrt {(7^{2} –4^{2})}\\\)

\(= 2 \times \sqrt{(49-16)}\\ = 2 \times 5.744\\\)

= 11.48 cm

Therefore, the chord length will be 11.48 cm

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