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A chord is a line segment whose two endpoints lie on a circle. A diameter is the special chord that passes through the center. Other chords do not need to pass through it.
Chords and subtend equal central angles, so they have equal length. Chord subtends a larger central angle and is longer.
Two chords of equal length have the same distance from the center of the circle. Conversely, two chords at the same distance from the center have equal length.
If , then the distance from center to chord equals the distance from to chord , that is .
A line drawn from the center of a circle perpendicular to a chord divides the chord into two equal parts.
In the figure above, and is the midpoint of chord , so .
To calculate the length of a chord, we can use the formula:
Here:
As the smaller central angle increases from to , the factor increases from to . The chord length therefore increases from to the diameter .
The distance of a chord from the center of the circle can be calculated using the formula:
Or if the chord length is known:
If two chords intersect inside a circle, then the product of the segments of one chord equals the product of the segments of the other chord.
In the figure above, the following holds:
Inscribed angles that subtend the same chord have equal measures when their vertices lie on the same side of that chord.
In the figure above, because both vertices lie on the same side of chord and both angles subtend that chord.
The apothem of a chord is the perpendicular segment from the center of the circle to the chord. Its length is the shortest distance between the center and the chord.
The length of the apothem can be calculated using the formula:
Here:
Two parallel chords in a circle have special properties.
If , then arc equals arc .
A circle has a radius of . If the central angle subtending a chord is , determine:
Two chords and intersect at point inside a circle. If , , and , find the length of .
In a circle with radius , there is a chord of length . Calculate the distance of this chord from the center of the circle.
Two parallel chords in a circle are and away from the center respectively. If the radius of the circle is , determine the lengths of both chords.
Prove that the longest chord in a circle is the diameter.
Calculating chord length and its distance from center
Given: , rad
Intersecting chords theorem
Given: , ,
Calculating distance of chord from center
Given: ,
Parallel chords
Given: , ,
Proof that diameter is the longest chord
For any chord, take to be the smaller central angle, so :
Published: . Updated: .
For the first chord:
For the second chord:
The maximum value of is achieved when , that is .
When , the chord passes through the center of the circle (diameter) with length:
Therefore, the diameter is the longest chord.
