NCERT Solutions Class 9 Maths Chapter 9 helps you understand Circles through detailed solutions covering angle subtended by chords, perpendicular from center to chord, and equal chords equidistant from center. You’ll learn how to prove circle theorems, apply angle properties in cyclic quadrilaterals, and solve problems involving chord lengths and arc measures. These geometric concepts are essential for Class 10 coordinate geometry and trigonometry.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 9 Maths Chapter 9 Circles – Complete Guide
NCERT Class 9 Chapter 9 – Circles introduces you to one of the most elegant and important shapes in geometry. You’ll dive deep into the properties of circles, starting with basic definitions and progressing to significant theorems about chords, angles subtended by arcs, and cyclic quadrilaterals. This chapter builds upon your understanding of angles and triangles from previous chapters, creating a strong foundation for advanced geometry in Class 10.
📊 CBSE Class 9 Maths Chapter 9 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 4 Marks |
| Unit Name | Geometry |
| Difficulty Level | Medium |
| Importance | Medium |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 1-2 questions |
You’ll explore fascinating relationships within circles, such as how equal chords are equidistant from the center, how angles in the same segment are equal, and the special properties of cyclic quadrilaterals. These concepts aren’t just theoretical—circles appear everywhere in real life, from wheels and gears to planetary orbits and architectural designs. Understanding circular geometry helps you appreciate the mathematical precision behind these structures.
For CBSE board examinations, this chapter carries 4 marks weightage with medium difficulty level. You can expect questions ranging from 2-mark theorem-based problems to 3-mark application questions involving construction and proof. The chapter typically includes both theoretical questions asking you to prove theorems and practical problems requiring you to apply these theorems to find angles, lengths, or prove geometric relationships.
Quick Facts – Class 9 Chapter 9
| 📖 Chapter Number | Chapter 9 |
| 📚 Chapter Name | Circles |
| ✏️ Total Exercises | 3 Exercises |
| ❓ Total Questions | 15 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering Circles is crucial not only for your Class 9 exams but also for Class 10, where you’ll encounter advanced topics like tangents and areas related to circles. The logical reasoning and proof-writing skills you develop here will serve you throughout your mathematical journey. With consistent practice of NCERT solutions and understanding each theorem’s application, you’ll confidently tackle any circle-related problem in your examinations.
NCERT Solutions Class 9 Maths Chapter 9 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| EXERCISE 9.1 | Complete step-by-step solutions for 2 questions | 📥 Download PDF |
| EXERCISE 9.2 | Complete step-by-step solutions for 6 questions | 📥 Download PDF |
| EXERCISE 9.3 | Complete step-by-step solutions for 7 questions | 📥 Download PDF |
Circles – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Angle Subtended by a Chord at the Centre \[\angle AOB = 2 \angle ACB\] | The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Note: A, B are points on the circle, O is the centre, and C is a point on the remaining part of the circle. | To find the angle at the centre or at any point on the circle if one of them is given. |
| Angles in the Same Segment \(\angle ACB = \angle ADB\) | Angles in the same segment of a circle are equal. Note: A, B, C, and D are points on the circle, and C and D lie on the same side of the chord AB. | To prove that two angles are equal, or to find an unknown angle if another angle in the same segment is known. |
| Angle in a Semicircle \(\angle ACB = 90^\circ\) | Angle in a semicircle is a right angle. Note: AB is the diameter, and C is any point on the circle. | When you have a diameter and an angle formed on the circle by the diameter’s endpoints. Useful for proving right angles. |
| Cyclic Quadrilateral Property 1 \(\angle A + \angle C = 180^\circ\) and \(\angle B + \angle D = 180^\circ\) | Opposite angles of a cyclic quadrilateral are supplementary (add up to 180 degrees). Note: A, B, C, and D are vertices of the quadrilateral lying on the circle. | When dealing with cyclic quadrilaterals. To find unknown angles if the opposite angle is known. To prove that a quadrilateral is cyclic. |
| Cyclic Quadrilateral Property 2 (Exterior Angle) \(\angle CBE = \angle ADC\) | If a side of a cyclic quadrilateral is produced, then the exterior angle is equal to the interior opposite angle. Note: ABCD is a cyclic quadrilateral, and side AB is extended to E. | When you have a cyclic quadrilateral with an extended side. Useful for finding exterior angles. |
| Equal Chords and Equal Angles If \(AB = CD\), then \(\angle AOB = \angle COD\) | Equal chords of a circle (or congruent circles) subtend equal angles at the centre. Note: O is the centre of the circle, and AB and CD are equal chords. | To prove that two angles are equal, given that the chords subtending them are equal. Or, to prove that two chords are equal if they subtend equal angles at the centre. |
| Equal Chords and Distance from Centre If \(AB = CD\), then \(OL = OM\) | Equal chords of a circle (or congruent circles) are equidistant from the centre. Note: O is the centre of the circle, AB and CD are equal chords, OL is perpendicular to AB, and OM is perpendicular to CD. | To prove that two chords are equidistant from the centre, given that they are equal. Or, to prove that two chords are equal if they are equidistant from the centre. |
| Perpendicular from Centre to Chord If \(OL \perp AB\), then \(AL = LB\) | The perpendicular from the centre of a circle to a chord bisects the chord. Note: O is the centre of the circle, AB is a chord, and OL is perpendicular to AB. | When you have a perpendicular line from the center of the circle to a chord. Useful for finding lengths of chord segments. |
| Line Through Centre to Midpoint of Chord If \(AL = LB\), then \(OL \perp AB\) | The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord. Note: O is the centre of the circle, AB is a chord, L is the midpoint of AB. | To prove that a line from the center is perpendicular to a chord, given that it passes through the midpoint of the chord. |
| One and only one circle passing through 3 points N/A | There is one and only one circle passing through three given non-collinear points. Note: Important to know this is only true for non-collinear points. Three collinear points will not form a circle. | Conceptual questions; when dealing with construction. |
Frequently Asked Questions – NCERT Class 9 Maths Chapter 9
📚 Related Study Materials – Class 9 Maths Resources
| Resource | Access |
|---|---|
| NCERT Class 9 Mathematics Textbook | Download Book |
| NCERT Class 9 Science Solutions | View Solutions |
| RD Sharma Class 9 (Updated 2025-26) | View Solutions |
| NCERT Class 9 English (Beehive) | Download Book |