Mathématiques
Chord of a Circle: Where It Sits Among the Other Circle Words
A chord joins two points on a circle's circumference, and the diameter is the longest one. Plus the perpendicular-from-the-centre rule, worked through.
Chord
A chord is a straight line joining any two points on the circumference of a circle; the longest chord in a circle is its diameter.
- Où les élèves le rencontrent
- Grade 8 mathematics, when the parts of a circle are named in full, and again at GCSE Higher tier in circle theorems.
La réponse en bref
A chord is a straight line joining two points on the circumference of a circle. The longest chord in any circle is its diameter, which is the chord that passes through the centre. A perpendicular line drawn from the centre to a chord always cuts that chord exactly in half.
Un exemple
√(13² − 5²) = √144 = 12, so the chord is 24 cm
A circle has radius 13 cm and a chord whose perpendicular distance from the centre is 5 cm. Drawing the radius to one end of the chord makes a right-angled triangle with hypotenuse 13 and one side 5, so the remaining side is 12 cm. That 12 cm is only half the chord, because the perpendicular bisects it, and the full chord measures 24 cm. Forgetting to double is the usual slip.
Five lines on one circle, and what separates them
The words are only confusing because they are usually met all at once. Drawn on a single circle they separate cleanly: some are straight, some are curved, some touch the centre and some never go near it.
Primary school names only the radius, diameter and circumference. Chord, tangent, arc, sector and segment arrive later, which is why a Grade 8 student can feel behind on vocabulary they were never actually taught.
- Radius — a straight line from the centre to the circumference; it is not a chord, because only one end sits on the circle
- Chord — a straight line from one point on the circumference to another
- Diameter — the chord that happens to pass through the centre, and therefore the longest chord there is
- Tangent — a straight line touching the circle at exactly one point, meeting the radius at that point at a right angle
- Arc — part of the circumference itself, so it is curved rather than straight
The perpendicular from the centre bisects the chord
Drop a line from the centre so that it meets a chord at right angles, and it will always cut the chord into two equal halves. The reason is symmetry: the two radii drawn to the ends of the chord are the same length, so the figure folds neatly along that perpendicular.
This is the fact that makes chord questions answerable, because it manufactures a right-angled triangle out of a radius, half the chord and the distance from the centre. Once that triangle exists, Pythagoras finds whichever of the three is missing.
What a chord creates once it is drawn
A chord divides the circle into two regions called segments — the smaller one is the minor segment, the larger the major segment. It also divides the circumference into two arcs, one on each side of it.
Keeping the three words straight is mostly a matter of what kind of object each one is. The chord is a line, an arc is a curve, and a segment is a region of the circle with an area. A question asking for the perimeter of a segment wants the chord plus the arc, not the chord twice.
Questions fréquentes
Is a diameter a chord?
Yes, and it is the longest one. A diameter joins two points on the circumference like any chord, but it also passes through the centre, which is what makes it maximal. Every other chord is shorter and sits at some positive distance from the centre.
Is a radius a chord?
No. A chord needs both of its ends on the circumference, and a radius has one end at the centre. The distinction matters in circle-theorem questions, where a radius drawn to the end of a chord is the standard construction precisely because the two are different objects.
How do you find the length of a chord?
Join the centre to the chord at a right angle, then to one end of the chord. That gives a right-angled triangle with the radius as its hypotenuse. Pythagoras returns half the chord, so double the result. With radius 10 cm and a distance of 6 cm, half the chord is 8 cm and the chord is 16 cm.
What is the difference between a chord and an arc?
A chord is the straight route between two points on the circle; the arc is the curved route along the circumference between the same two points. The arc is always the longer of the two. Together they enclose a segment, which is the region trapped between them.
Sources
- National curriculum in England: mathematics programmes of study — Department for Education
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