Chord of Circle

The chord of a circle is 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 that passes through the center of the circle.

A line segment that joins two points on the circumference of the circle is defined as the chord of the circle. Among the other line segments that can be drawn in a circle, the chord is one whose endpoints lie on the circumference. Observe the following circle to identify the chord PQ. Diameter is also considered to be a chord which passes through the center of the circle.

Given below are a few important properties of the chords of a circle.

• The perpendicular to a chord, drawn from the center of the circle, bisects the chord.
• Chords of a circle, equidistant from the center of the circle are equal.
• There is one and only one circle which passes through three collinear points.
• When a chord of 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.
• A chord when extended infinitely on both sides becomes a secant.

There are two basic formulas to find the length of the chord of a circle:

• Chord length using perpendicular distance from the center = 2 × √(r² − d²). Let us see the proof and derivation of this formula. In the circle given below, radius 'r' is the hypotenuse of the triangle that is formed. Perpendicular bisector 'd' is one of the legs of the right triangle. We know that the perpendicular bisector from the center of the circle to the chord bisects the chord. Therefore, half of the chord forms the other leg of the right triangle. By Pythagoras theorem, (1/2 chord)² + d² = r², which further gives 1/2 of Chord length = √(r² − d²). Thus, chord length = 2 × √(r² − d²)

• Chord length using trigonometry = 2 × r × sin(θ/2); where 'r' is the radius of the circle and 'θ' is the angle subtended at the center by the chord. Observe the following circle to see the central angle 'θ' subtended by the chord AB and 'r' as the radius of the circle.
  

The chord of a circle has a few theorems related to it.

Theorem 1: The perpendicular to a chord, drawn from the center of the circle, bisects the chord.

Observe the following circle to understand the theorem in which OP is the perpendicular bisector of chord AB and the chord gets bisected into AP and PB. This means AP = PB

Theorem 2: Chords of a circle, equidistant from the center of the circle are equal.

Observe the following circle to understand the theorem in which chord AB = chord CD, and they are equidistant from the center if PO = OQ.

Theorem 3: For two unequal chords of a circle, the larger chord will be closer to the center than the smaller chord. (Unequal Chords Theorem)

If we draw multiple chords in a circle starting from the diameter to both the ends, we will observe that as we move closer to the center, the chord increases in length.

Important Notes

• The radius of a circle bisects the chord at 90°.
• When two radii join the two ends of a chord, they form an isosceles triangle.
• The diameter is the longest chord of a circle.

Related Links:

• Circles
• Area of a Circle
• Area of a Sector of a Circle
• Segment of a Circle
• Arc Length

What is the Chord of a Circle in Mathematics?

The chord of a circle refers to a straight line joining two points on the circumference of the circle. The longest chord in a circle is its diameter which passes through its center.

How to Find the Chord of Circle?

Any line segment whose endpoints are on the circumference of the circle is the chord of that circle. The length of the chord of a circle can be calculated according to the given dimensions, with the help of two methods.

• When the radius and the distance from the center of the circle to the chord is given, we need to apply the chord length formula: Chord length = 2√(r²-d²); where 'r' is the radius of the circle and 'd' is the perpendicular distance from the center of the circle to the chord.
• When the radius and the central angle is given, we need to apply the formula, Chord length = 2 × r × Sin (θ/2); where 'r' is the radius and 'θ' is the central angle subtended by the chord.

What is the Longest Chord of a Circle?

The longest chord of a circle is its diameter. It is the line that passes through the center of a circle touching two points on the circumference.

How to Find the Length of Chord of Circle with a Given Radius?

If the radius and the distance of the center of the circle to the chord are given, the chord of the circle can be calculated. We just need to apply the chord length formula: Chord length = 2√(r²-d²), where 'r' is the radius of the circle and 'd' is the perpendicular distance from the center of the circle to the chord.

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. In other words, a line drawn through the center of a circle to bisect a chord is perpendicular to the chord.

What is the Radius, Diameter, and Chord of a Circle?

The radius of a circle is the distance from the center to any point on the circumference. Diameter is the line segment that passes through the center of a circle touching two points on the circumference of the circle. A chord is a line segment that joins any two points on the circumference of the circle.

How to Draw the Chord of a Circle?

The chord of a circle can be constructed with the help of a compass and a ruler. For example, let us draw a chord of length 4 inches in a circle that has a radius of 2 inches.

• Mark the center of the circle as O and using a compass draw a circle with a radius of 2 inches.
• Then, mark a point C on the circumference of the circle, and with C as the center draw an arc that cuts the circumference at another point D.
• Join C and D. CD is the required chord that measures 4 inches.