Chords of a Circle
The chord of a circle can be defined as the line segment joining any two points on the circumference of the circle and thus we can say that the diameter is the longest chord of a circle that passes through the center of the circle.
Chords of a Circle Definition
A line segment that joins two points on the circumference of the circle is defined to be the chord of a circle. In the given circle with ‘O’ as the center, AB represents the chord of the circle while the longest chord will pass through the center O, which is the diameter of the circle.
Equal and Unequal Chords in a Circle
Chords that are equidistant from the center of a circle are considered equal chords. On the other hand, the chords that are not equidistant from the center of a circle are considered unequal chords.
Properties of Chords of a Circle
Given below are a few important properties of chords of a circle.
 When a chord of a circle is drawn, it divides the circle into two regions, referred to as the segments of the circle: the major segment and the minor segment.
 Arc length is the distance between two points along the boundary of the circle. Thus, we can say that the length of the chord belonging to the greater arc length is greater.
 A chord when extended infinitely on both sides becomes secant.
Chords of a Circle Formula
There are two basic formulas to find the length of the chord of a circle:
 Chord Length Using Perpendicular Distance from the Center = 2 × √(r^{2} − d^{2}). Proof: In the circle given below, radius r will be the hypotenuse of the triangle so formed, Perpendicular bisector d will be one of the legs of the right angle. Since we know that the perpendicular bisector from the chord to the center of the circle bisects the chord. Thus half of the chord forms the other leg of the right triangle. By Pythagoras theorem, (1/2 chord)^{2 }+ d^{2 }= r^{2 } which further gives 1/2 of Chord length = √(r^{2} − d^{2}). Thus, chord length = 2 × √(r^{2} − d^{2})
 Chord Length Using Trigonometry = 2 × r × sin(c/2)
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.
Chords of a Circle Theorems
 Equal chords of a circle are equidistant from the center. (Equal Chords Theorem)
 Two chords of a circle that is equidistant from its center must have the same length. (Converse of Equal Chords Theorem)
 For two unequal chords of a circle, the larger chord will be closer to the center than the smaller chord. (Unequal Chords Theorem)
Visit Equal and Unequal Chords page for proof of these theorems.
Relationship Between Radius and Chords of a Circle
The radius of a circle is any line segment that connects the center of the circle to any point on the circle whereas the chord of a circle is a line segment joining any two points on the circumference of the circle. The diameter which is twice the radius is the chord of a circle that passes through the center of the circle.
Important Notes
 A line drawn from the center of a circle to bisect the chord is perpendicular to the chord is referred to as the perpendicular bisector of that chord.
 While comparing the length of two arcs, the length of the chord that belongs to the greater arc length is greater.
Challenging Question
A circle has a radius of 25 inches. Find the distance between the two parallel chords which are 48 inches and 40 inches in length if they lie on
 opposite sides of the center.
 same side of the center.
Topics Related to Chords of a Circle
Solved Examples on Chords of a Circle

Example 1: In the given circle, O is the center, and chord AB is 16 cm. Find the length of AD if OM is the radius of the circle.
Solution:
We know that the radius of a circle is always perpendicular to the chord and acts as a perpendicular bisector. Therefore, \(\mathrm{AD}=\frac{1}{2} \times \mathrm{AB}\) = 16/2 = 8. Therefore, AD = 8 cm.

Example 2: In the given circle, O is the center with a radius of 5 cm. Find the length of chord AB if the length of the perpendicular drawn from the center is 4 cm.
Solution:
Since OM is perpendicular to AB, △AOM will be a rightangled triangle. In △AOM, from Pythagoras theorem, OA^{2} = OM^{2} + AM^{2}. After transposition of terms, we get AM^{2} = OA^{2}  OM^{2}. On substituting the values, AM^{2} = 5^{2}  4^{2}. Thus, AM = \(\sqrt{9}\) = 3. Therefore, Chord AB = 2 x 3= 6 cm
FAQs on Chords of a Circle
What is a Chord of a Circle?
A chord of a circle refers to a straight line joining two points on the circumference of the circle.
How do you Find the Chord of a Circle?
Any line segment whose endpoints are on the circumference of the circle is the chord of that circle.
What is the Longest Chord of a Circle?
The longest chord of a circle is its diameter, that is the line passing through its center.
How to Find the Length of a Chord of a Circle Given the Radius?
If the radius and the distance of the center of the circle to the chord are given, just apply the chord length formula: Chord length = 2√r^{2}d^{2} , where
 r is the radius of the circle
 d is the perpendicular distance of the center of the circle to the chord.
How are the Tangent Secant and Chord of a Circle Related?
A tangent is a line that intersects the curve or the circle at exactly one point. A secant is a line that intersects the curve at a minimum of two distinct points. A chord is a segment connecting two points on a circle. Thus, in a way chords become secants when extended.
What is the Relationship Between the Chord of a Circle and a Perpendicular to it from the Center?
The perpendicular drawn from the center of a circle to a chord bisects the chord or a line drawn through the center of a circle to bisect a chord is perpendicular to the chord.