CBSE Math RevisionCircles: CBSE Class 9 Maths knowledge organiser
Everything to know about circles on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.
Key definitions
- Chord
- A line segment joining two points on a circle.
- Arc
- A part of the circumference between two points.
- Cyclic quadrilateral
- A quadrilateral with all four vertices on one circle; its opposite angles add up to 180°.
Key formulas
| Chord \(2l\) at distance \(d\) from the centre | \(r^2=l^2+d^2\) |
| ⟂ from centre bisects a chord; equal chords ⇔ equidistant from centre ⇔ equal angles at centre | |
| Angle at centre = 2 × angle on the remaining arc; angles in the same segment are equal; angle in a semicircle = 90° | |
| Cyclic quadrilateral | \(\angle A+\angle C=\angle B+\angle D=180^\circ\) |
Worked example
A chord of a circle of radius \(17\) cm is \(30\) cm long. How far is it from the centre?
Half-chord \(15\), so \(d = \sqrt{17^2 - 15^2} = \sqrt{64} = 8\) cm.
Common mistakes
- Using the whole chord instead of half of it in \(r^2 = l^2 + d^2\).
- Halving the angle at the centre when the point is on the same arc (the minor arc) as the chord: then the angle is \(180^\circ\) minus half.
- Adding adjacent angles of a cyclic quadrilateral to \(180^\circ\): only opposite angles are supplementary.
- Two parallel chords: forgetting to check whether they are on the same side of the centre or on opposite sides.
You should be able to…
- Use the chord-distance relation and the basic angle facts: angle in a semicircle, angle at the centre, cyclic quadrilateral.
- Find chords, distances and angles in two- and three-mark questions and give the reason for each step.
- Combine the chord theorems in longer calculations and prove results about cyclic quadrilaterals.
- Handle parallel chords, intersecting circles and proofs such as 'a cyclic parallelogram is a rectangle'.
The printable sheet
Revise it next
- Circles: Class 9 notes
- Practise Circles by step (Route to 95)
- Skill Builders
- CBSE Class 9 Maths formula sheet (PDF)
Other CBSE Class 9 Maths topics: Number System · Introduction to Polynomials · Sequences and Progressions · Exploring Algebraic Identities · Linear Equations in Two Variables · Coordinate Geometry · Introduction to Euclid's Geometry: Axioms and Postulates · Lines and Angles · Triangles – Congruence Theorems · 4-gons (Quadrilaterals) · Area and Perimeter · Surface Area and Volume · Statistics · Introduction to Probability · All CBSE Class 9 Maths organisers