Chapter 9 · NCERT Class 9 Mathematics

Circles

A circle is every point at the same distance from one centre. From that single idea flow the radius, chord, diameter, arc and segment — and powerful rules about how chords and angles behave. Tap each term to see exactly what it points to, then test yourself.

📐 3 topics⏱ ~25 min📝 12-question quiz

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The language of circles

Every circle is built from the same handful of parts. Tap each term to see what it names and how the pieces relate to the centre.

Explore · Parts of a circletap a term

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The three big ideas

Centre & radius — the centre is the fixed point; the radius is the fixed distance from the centre to any point on the circle. All radii of one circle are equal.
Chord — a line segment joining any two points on the circle.
Diameter — a chord that passes through the centre; it is the longest chord and equals 2 × radius.
Arc — a part of the circle (a curved piece). The smaller is the minor arc, the larger the major arc.
Sector — the region between two radii and the arc between them (like a slice of pizza).
Segment — the region between a chord and the corresponding arc.

Common mistake: the diameter is the longest chord because it passes through the centre — not every chord, and the diameter always equals 2 × radius. A radius alone (centre to the boundary) is only half the diameter, so it is not a chord.

The perpendicular from the centre to a chord bisects the chord (cuts it into two equal halves).
Conversely, the line from the centre to the midpoint of a chord is perpendicular to that chord.
Equal chords of a circle are equidistant from the centre — and chords equidistant from the centre are equal.

The angle at the centre made by an arc/chord is twice the angle it makes at any point on the remaining circle.
Angles in the same segment (standing on the same arc) are equal.
In a cyclic quadrilateral (all four vertices on the circle), each pair of opposite angles adds up to 180°.

Worked example. A chord subtends 50° at the centre. What angle does it subtend at a point on the major arc?

Rule: angle at the centre = 2 × angle at a point on the remaining circle.

So: angle on the major arc = 50° ÷ 2 = 25°.

Where you'll meet it

Circles, at work

Wheels, gears & clocks

A wheel turns about its centre, so every point on the rim is one radius away. Gears mesh along their circumferences, and a clock's hands sweep equal angles about the centre — the angle-at-the-centre idea in everyday motion.

Circular structures

Domes, roundabouts and circular stadiums are laid out from a centre with a fixed radius. Engineers use chord and arc lengths to mark out the curve accurately on the ground.