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Circle problems look varied, but almost every Class 9 question uses one of about five facts. The difficulty is spotting which one, and drawing the extra line that makes it visible, usually a radius. This guide covers the theorems in the new chapter, four worked problems with numbers you can check, and the mistakes that cost marks.
What is in the Class 9 circles chapter now?
In NCERT's new textbook Ganita Manjari, circles are Chapter 5, "I'm Up and Down, and Round and Round". It proves twelve theorems, from "there is a unique circle passing through three non-collinear points" to the conditions for four points to lie on a circle. The CBSE Class IX curriculum for 2026-27 lists Circles in the Geometry unit (25 of the 80 theory marks), with outcomes such as constructing the circumcircle of a triangle and explaining why the angle in a semicircle is a right angle.
What do the chord theorems say?
Equal chords subtend equal angles at the centre, and chords that subtend equal angles at the centre are equal.
The line from the centre to the midpoint of a chord is perpendicular to it, and the perpendicular from the centre to a chord bisects it.
Equal chords are equally distant from the centre, and chords equally distant from the centre are equal.
Of two unequal chords, the longer one is closer to the centre. The diameter, through the centre, is the longest chord.
The working tool behind all of these: join the centre to the ends of the chord. You get an isosceles triangle (two radii), and the perpendicular from the centre splits it into two right triangles.
Worked example 1: chord length from distance
A circle has radius 13 cm. A chord is 5 cm from the centre. How long is the chord?
Draw the perpendicular from the centre O to the chord AB, meeting it at M. M is the midpoint of AB.
In right triangle OMA: OA = 13 (radius), OM = 5, so AM = square root of (169 − 25) = square root of 144 = 12 cm.
AB = 2 × 12 = 24 cm.
Worked example 2: two parallel chords
In a circle of radius 10 cm, two parallel chords have lengths 12 cm and 16 cm. How far apart are they?
Chord of 12 cm: half is 6, so its distance from the centre is the square root of (100 − 36) = 8 cm.
Chord of 16 cm: half is 8, so its distance is the square root of (100 − 64) = 6 cm.
If both chords are on the same side of the centre, they are 8 − 6 = 2 cm apart. If on opposite sides, 8 + 6 = 14 cm apart.
Questions often expect both answers. Check whether the question says which side.
What do the arc and angle theorems say?
The angle an arc subtends at the centre is double the angle it subtends at any point on the rest of the circle. Euclid proved this as Book III, Proposition 20.
Angles in the same segment are equal.
The angle in a semicircle is a right angle, because the angle at the centre is a straight angle of 180 degrees.
Opposite angles of a cyclic quadrilateral add up to 180 degrees, and conversely, if opposite angles of a quadrilateral add to 180, its vertices lie on a circle.
Worked example 3: angle at the centre
An arc subtends 110 degrees at the centre. What angle does it subtend at a point on the major arc (the rest of the circle)?
Half of 110 is 55 degrees. Common slip: doubling instead of halving. The angle at the centre is the bigger one.
Worked example 4: a cyclic quadrilateral
ABCD is cyclic, with angle B = (4x − 6) degrees and angle D = (2x + 18) degrees. Find both.
B and D are opposite angles, so B + D = 180.
4x − 6 + 2x + 18 = 180, so 6x + 12 = 180 and x = 28.
Angle B = 106 degrees, angle D = 74 degrees. Check: 106 + 74 = 180.
What mistakes should you avoid?
Halving and doubling the wrong way in the centre-angle theorem.
Using the 180-degree rule on adjacent angles. Only opposite angles of a cyclic quadrilateral add to 180.
Calling a quadrilateral cyclic without reason. You need four points on one circle given, or opposite angles proved supplementary.
Assuming any perpendicular bisects a chord. Only the perpendicular from the centre does.
A practice routine for circles
Testing yourself after a gap beats rereading. In Roediger and Karpicke (2006), students who took recall tests remembered more a week later than those who restudied.
List the theorems from memory and draw a small figure for each. Check against this post.
Two days later, do two chord-length problems and two angle problems without notes.
A week later, do a mixed set of six, writing the theorem name beside each step.
To see why the centre angle is always double as the point moves around the circle, try Fermi's Simulation Lab, which has interactive simulations for ideas that are easier to see than read. Cyclic quadrilaterals connect to quadrilaterals in Class 9, and tangents follow in circles and tangents in Class 10.
Draw a circle tonight, mark an arc, and measure the angle at the centre and at two points on the circle to see the 2:1 rule for yourself.
