Chord of a Circle, Its Length and Theorems: Chord of a Circle Definition
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Chord of a Circle Definition
The chord of a circle can be defined as the line segment joining any two points on the circumference of the circle. It should be noted that the diameter is the longest chord of a circle which passes through the center of the circle. The figure below depicts a circle and its chord.

Chord of a Circle Definition
In the given circle with ‘O’ as the center, AB represents the diameter of the circle (longest chord), ‘OE’ denotes the radius of the circle and CD represents a chord of the circle.
Let us consider the chord CD of the circle and two points P and Q anywhere on the circumference of the circle except the chord as shown. If the endpoints of the chord CD are joined to the point P, then the angle ∠CPD is known as the angle subtended by the chord CD at point P. The angle ∠CQD is the angle subtended by chord CD at Q. The angle ∠COD is the angle subtended by chord CD at the center O.

Chord of a Circle Definition
Angle Subtended by Chord
Chord Length Formula
There are two basic formulas to find the length of the chord of a circle which are:
Formula to Calculate Length of a Chord | |
Chord Length using Perpendicular Distance from the Centre | Chord Length |
Chord Length using trigonometry | Chord Length |

Chord Length Formula
Chord Length of a Circle Formula
Where,
r is the radius of the circle
c is the angle subtended at the center by the chord
d is the perpendicular distance from the chord to the circle center
Chord of a Circle Theorems
If we try to establish a relationship between different chords and the angle subtended by them on the center of the circle, we see that the longer chord subtends a greater angle at the center. Similarly, two chords of equal length subtend equal angle at the center. Let us try to prove this statement.
Theorem 1: Equal Chords Equal Angles Theorem
Statement: Chords which are equal in length subtend equal angles at the center of the circle.

Equal Chords Equal Angles
Equal Chords Equal Angles Theorem
Proof:
From fig. 3, In and
Sr. No. | Statement | Reason |
1. | Chords of equal length (Given) | |
2. | Radius of the same circle | |
3. | SSS axiom of Congruence | |
4. | By CPCT from statement 3 |
Note: CPCT stands for congruent parts of congruent triangles.
The converse of theorem 1 also holds true, which states that if two angles subtended by two chords at the center are equal then the chords are of equal length. From fig. 3, if , then . Let us try to prove this statement.
Theorem 2: Equal Angles Equal Chords Theorem (Converse of Theorem 1)
Statement: If the angles subtended by the chords of a circle are equal in measure then the length of the chords is equal.

Equal Angles Equal Chords
Equal Angles Equal Chords Theorem
Proof:
From fig. 4, In and
Sr. No. | Statement | Reason |
1. | Equal angle subtended at centre O (Given) | |
2. | Radii of the same circle | |
3. | SAS axiom of Congruence | |
4. | From Statement 3 (CPCT) |
Theorem 3: Equal Chords Equidistant from Center Theorem
Statement: Equal chords of a circle are equidistant from the center of the circle.
Proof:
Given: Chords AB and CD are equal in length.
Construction: Join A and C with centre O and drop perpendiculars from O to the chords AB and CD.

Equal Chords Equidistant
Equal Chords Equidistant from Center Theorem
S. No. | Statement | Reason |
1. | , | The perpendicular from centre bisect the chord |
In and | ||
2. | and | |
3. | Radii of the same circle | |
4. | Given | |
5. | R.H.S. Axiom of Congruency | |
6. | Corresponding parts of congruent triangle | |
7. | From statement (1) and (6) |
Frequently Asked Questions
What is a Circle?
A circle is defined as a closed two-dimensional figure who’s all the points in the boundary are equidistant from a single point called its centre.
What is the Chord of a Circle?
The chord is a line segment that joins two points on the circumference of the circle. A chord only covers the part inside the circle.
What is the Formula of Chord Length?
The length of any chord can be calculated using the following formula:
Chord Length
Is Diameter a Chord of a Circle?
Yes, diameter is also considered as a chord of the circle. The diameter is the longest chord possible in a circle and it divides the circle into two equal segments.
Example Question Using Chord Length Formula
Question: Find the length of the chord of a circle where the radius is 7 cm and perpendicular distance from the chord to the center is 4 cm?
Solution:
Given radius,
and distance,
Formula for the chord length of a circle,
Chord length
Put the value of diameter and radius.
Chord length
Put the value of square of 7 and 4.
Chord length
Subtract 16 from the 49.
Chord length
Chord length
Chord length