A chord is a line segment whose two endpoints both lie on a circle. The diameter is the longest chord, because it passes through the center. A chord at distance d from the center of a circle with radius r has length 2√r2 − d2, so chords closer to the center are longer.
In Spanish
cuerda
Also called
chord of a circle, circle chord
How to say it
KORD
Grade level
Geometry
Grade 7GeometryCCSS HSG-C.A.2SATACTGED
Chord in a picture
The radius, the distance d and half the chord make a right triangle, which is where the chord length formula comes from.
Formal definition and notation
A chord of a circle is a segment AB where A and B are distinct points on the circle. Every chord lies inside the circle. The perpendicular from the center to a chord bisects the chord, and a chord at distance d from the center of a circle of radiusr has length2√r2 − d2 for 0 ≤ d < r. Equivalently, a chord that subtends a central angleθ has length 2rsin ( θ2 ). More generally, a chord of any curve is a segment joining two of its points.
Symbol
Read it as
Meaning
AB
segment A B
the chord with endpoints A and B on the circle
AB
A B
the length of that chord
Key facts
Both endpoints must be on the circle. A segment from the center to the circle is a radius, not a chord.
Every diameter is a chord, but most chords are not diameters. A diameter is the only chord through the center.
A line drawn from the center perpendicular to a chord cuts the chord into two equal halves.
Chords that are the same distance from the center have the same length, and the closer a chord is to the center, the longer it is.
A chord cuts a circle into two arcs; a secant is the full line that contains a chord.
Chord length formula
Chord length formula
chord = 2√r2 − d2
r
the radius of the circle
d
the perpendicular distance from the center to the chord
It comes from the Pythagorean theorem: the radius, the distance d and half the chord form a right triangle, so half the chord is √r2 − d2. When d = 0, the chord is a diameter, 2r.
Chord examples, step by step
Example 1: The longest chord
Warm-up
A circle has a radius of 9 cm. How long is its longest chord?
The longest chord passes through the center: it is a diameter.
A diameter is twice the radius: 2 × 9 = 18.
No other chord can be longer, because every other chord is closer to the edge.
Answer: The longest chord is 18 cm.
Example 2: Chord length from the distance to the center
Core
A circle has radius 10 in. A chord is 6 in from the center. How long is the chord?
A diameter is a special chord that passes through the center; any segment with both ends on the circle is a chord.
Chordvs.Secant and tangent
A secant is a whole line that crosses a circle at two points (it contains a chord). A tangent is a line that touches the circle at exactly one point.
Chordvs.Arc
An arc is the curved part of the circle between two points; the chord is the straight segment joining the same two points.
Quick check: 3 questions
Question 1 of 3
Which of these is always a chord of a circle?
Show answer and explanation
Answer: B. A chord joins two points on the circle. The first choice is a radius, the third is a tangent and the fourth is an arc.
Question 2 of 3
A circle has radius 5. A chord is 3 units from the center. How long is the chord?
Show answer and explanation
Answer: C. Half the chord is √52 − 32 = 4, so the chord is 8. 4 forgets to double, and 2√34 adds the squares instead of subtracting.
Question 3 of 3
Which statement is true?
Show answer and explanation
Answer: C. A diameter has both endpoints on the circle, so it is a chord, and it is the longest one. Most chords miss the center, a radius has only one endpoint on the circle, and no chord is longer than the diameter.
Chord: frequently asked questions
Is the diameter a chord?
Yes. A diameter has both endpoints on the circle, so it meets the definition of a chord. It is the longest chord because it is the only one that passes through the center. Every other chord is shorter than the diameter, and a radius is not a chord because only one of its endpoints is on the circle.
How do you find the length of a chord?
If you know the radius r and the distance d from the center to the chord, use 2√r2 − d2. For r = 10 and d = 6, the chord is 2√64 = 16. If you know the central angle θ instead, the chord is 2rsin ( θ2 ).
What is the difference between a chord and a secant?
A chord is a segment: it starts and stops on the circle. A secant is a line that passes through the circle at two points and keeps going in both directions. Every secant contains exactly one chord, the part of it that lies inside the circle.
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