What is Chord Length Formula?

There are two basic methods or formulas to calculate the length of the chord. a chord length can be determined by using the perpendicular distance from the centre of the circle as well as by the trigonometric method. Thus the length of a chord can be found

• Using the Pythagorean Theorem
• Using the Law of Cosines

Let’s understand these methods in detail as follows:

Method 1: Using the Pythagorean Theorem

In the following diagram for a chord, as we know the perpendicular drawn from the centre of the circle to the chord bisects it in two halves.

In triangles OAM, using Pythagoras Theorem,

r² = x² + d²

⇒ x² = r² – d²

⇒ x = √(r² – d²)

As x is half the length of the chord,

Thus, the chord length for any circle with its perpendicular distance from the centre is known is given as

Length of a Chord of a Circle = 2 ×[√(r² – d²)]

Where,

• r is the radius of circle, and 
• d is the perpendicular distance between center of circle and chord.

Method 2: Using the Law of Cosines

As we know for a triangle ABC, with sides a, b and c, the Law of cosine states,

c² = a² + b² – 2ab cos C

Using, this law in the following diagram of a chord subtending θ angle at the centre of the circle, we can find the length of the chord.

In triangle OAB, using the Law of cosine,

⇒ x² = r² + r² – 2×r×r×cos θ

⇒ x² = 2r²– 2r²cos θ

⇒ x² = 2r²(1- cos θ)

⇒ x =

Thus, the Chord length is given by:

Chord Length = 2r × sin [θ/2]

Where, 

• θ is the angle subtended by the chord at the center, and 
• r is the radius of the circle.

Other Related Formula for Chord Length

When two circles share a common chord, then the length of that common chord can be calculated using the formula

Length of a Common Chord of Two Circles = 2R₁ × R₂ / D

Where,

• R₁ and R₂ refers to radius of circles
• D is the distance between the two centers of the circle

Chord of a circle is the line that joints any two points on the circumference of the circle. A circle can have various chords and the largest chord of a circle is the diameter of the circle. We can easily calculate the length of the chord using the Chord Length Formula. As the name suggests it is the formula for calculating the length of the chord in a circle in Geometry.

In this article, we will learn about the definition of the chord, theorems of the chords and the circle, explain its properties, and the formulas to calculate the length of the chord using different methods. The article also has some solved sample problems for better understanding.