How to Find a Chord Length from Its Distance to the Centre

What is the chord-length formula?

In a circle of radius r, a chord at perpendicular distance d from the centre has length 2√(r² − d²). The perpendicular from the centre bisects the chord, so Pythagoras first gives half its length. Use consistent units and require 0 ≤ d ≤ r.

Build the right triangle

Call the centre O, the chord AB and the foot of the perpendicular M. Then AM = MB, OM = d and OA = r. Triangle OMA is right-angled at M. Therefore AM² + d² = r², giving AM = √(r² − d²). Double AM to obtain the complete chord AB.

The radius is the hypotenuse. Putting the chord length in that position is a common diagram error. Draw the radius to a chord endpoint rather than to its midpoint.

Worked example

A circle has radius 13 cm and its chord lies 5 cm from the centre. Half the chord is √(169 − 25) = √144 = 12 cm. The full chord is therefore 24 cm. As a check, a chord cannot be longer than the diameter, which is 26 cm here.

Reverse the question

A chord is 16 cm long in a circle of radius 10 cm. Half its length is 8 cm, so the perpendicular distance is √(100 − 64) = 6 cm. Using the full 16 inside Pythagoras would give a negative quantity under the square root, signalling the wrong triangle.

Boundary cases and practice

If d = 0, the chord goes through the centre and has length 2r: it is a diameter. As d increases towards r, chord length decreases. At d = r, the intersection has collapsed to one point, not a positive-length chord.

• r = 5 cm, d = 3 cm: chord = 8 cm.
• r = 17 cm, d = 8 cm: chord = 30 cm.
• r = 4 cm, d = 6 cm: impossible for a chord of this circle.

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