For AI agents: use /llms.txt for the Nakafa content index.
A chord is a line segment that connects two points on a circle. Unlike a diameter that passes through the center of the circle, a chord can be positioned anywhere as long as both endpoints lie on the circle.
In the figure above, , , and are chords with different lengths.
Two chords of equal length have the same distance from the center of the circle.
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:
Where:
The relationship between central angle and chord length can be visualized as follows:
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.
In the figure above, because both subtend the same chord .
An apothem is the shortest distance from the center of a circle to a chord, which is the perpendicular line from the center to the chord.
The length of the apothem can be calculated using the formula:
Where:
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 with central angle :
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.
