Chord–Distance from Centre Relationship

The Chord–Distance Theorem explains how the distance of a chord from the centre of a circle affects its length.

It states that:

In the same circle (or in congruent circles), a chord that is closer to the centre is longer, and a chord that is farther from the centre is shorter.

The distance of a chord from the centre is the length of the perpendicular dropped from the centre to the chord.

If \(O\) is the centre and \(AB\) is a chord, then the perpendicular from \(O\) meets the chord at \(M\), and:

OM = \text{distance of chord } AB \text{ from centre}

The length of a chord depends on the radius \(r\) of the circle and the perpendicular distance \(d\) from the centre.

\(\text{Chord Length} = 2 \sqrt{r^2 - d^2}\)

If two chords are at distances \(d_1\) and \(d_2\) from the centre:

The diameter is the chord with the minimum distance from the centre — distance = 0.

So, it becomes the longest chord in the circle.

In a circle of radius \(10\text{ cm}\), a chord is \(6\text{ cm}\) away from the centre.

Then its length is:

AB = 2\sqrt{10^2 - 6^2} = 2\sqrt{100 - 36} = 2\sqrt{64} = 16\text{ cm}

Imagine moving a chord up and down inside a circle:

• When the chord is close to the centre, it looks longer.
• As the chord moves toward the boundary, it becomes shorter.
• At the centre (distance = 0), it becomes a diameter.

This concept is used in:

• Designing wheels and rims
• Engineering circular frames
• Locating the centre using multiple chords
• Understanding tension and shape symmetry in circular objects