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CBSE • Class 9 • Mathematics • Ch 9
Estimated Time: 45 Mins
Study Progress: In Progress

Circles

In Class 9 Mathematics, Chapter 9 "Circles" investigates the geometry of circles, chords, subtended angles, cyclic quadrilaterals, and fundamental circle theorems aligned with the 2026–27 NCERT syllabus.

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🎡 Have You Ever Wondered?

Why is the wheel universally hailed as humanity's greatest mechanical invention?

A circle is the only shape where every single point on its perimeter sits at the exact same distance from the center. This mathematical symmetry guarantees that when a wagon rolls on circular wheels, its axle remains at a perfectly constant height, preventing the violent jarring of square or triangular wheels!

From Ferris wheels to planetary orbits, circles are nature's ultimate symmetry. In this chapter, you will master the chord and arc theorems of circles.

Why This Chapter Matters

In Class 9 Mathematics, Chapter 9 "Circles" investigates the geometry of circles, chords, subtended angles, cyclic quadrilaterals, and fundamental circle theorems aligned with the 2026–27 NCERT syllabus.

Before You Begin (Prerequisites)

  • Circle terminology: Radius, diameter, chord, arc, sector, and segment.
  • Angle sum property and triangle congruence from Chapters 6 & 7.

What You Will Learn (Core Objectives)

  • Prove that the perpendicular from the center of a circle to a chord bisects the chord.
  • Prove that equal chords of a circle subtend equal angles at the center (and converse).
  • Prove that equal chords are equidistant from the center.
  • Prove the Central Angle Theorem: Angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
  • Prove that the angle in a semicircle is a right angle ($90^\circ$).
  • Prove that opposite angles of a Cyclic Quadrilateral sum to $180^\circ$.

Chapter Roadmap & Progression

1 1. Chords & Their Distance from the...
2 2. Angle Subtended by an Arc & Cycl...

Complete Concept Guide (100% Curriculum Coverage)

1. Chords & Their Distance from the Center

1. Core Theorems
  • Theorem 9.1: Equal chords of a circle subtend equal angles at the center.
  • Theorem 9.2 (Converse): If angles subtended by chords at the center are equal, the chords are equal.
  • Theorem 9.3: The perpendicular from the center of a circle to a chord bisects the chord ($OM \perp AB \implies AM = MB$).
  • Theorem 9.4 (Converse): The line drawn through the center of a circle to bisect a chord is perpendicular to the chord.
  • Theorem 9.5: Equal chords of a circle are equidistant from the center.

2. Angle Subtended by an Arc & Cyclic Quadrilaterals

1. The Double Angle Theorem

Theorem 9.7 (Central Angle Theorem): The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle:

$$\mathbf{\angle AOB = 2 \angle APB}$$

  • Corollary 1: Angles in the same segment of a circle are equal ($\angle APB = \angle AQB$).
  • Corollary 2: The angle in a semicircle is a right angle ($90^\circ$).
  • Theorem 9.10 (Cyclic Quadrilateral): A quadrilateral whose all four vertices lie on a circle has supplementary opposite angles: $$\mathbf{\angle A + \angle C = 180^\circ \quad \text{and} \quad \angle B + \angle D = 180^\circ}$$

Central Angle & Cyclic Quadrilateral Geometry Matrix

Central Angle & Cyclic Quadrilateral Theorems Theorem 9.7 (Angle at center = 2 x angle at circumference) and cyclic properties O A B P x 2x Angle AOB = 2 x Angle APB A B C D Angle A + Angle C = 180 deg

Chapter Summary & 10 Key Takeaways

Takeaway 1
Chord Perpendicular: The perpendicular from the center of a circle to a chord bisects the chord.
Takeaway 2
Equal Chords: Subtend equal angles at the center and are equidistant from the center.
Takeaway 3
Central Angle Theorem: Angle at the center is double the angle subtended at any point on the remaining circumference.
Takeaway 4
Semicircle Angle: The angle subtended by a diameter in a semicircle is always a right angle ($90^\circ$).
Takeaway 5
Cyclic Quadrilateral: A quadrilateral whose all 4 vertices lie on a circle; opposite angles sum to $180^\circ$.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
A chord of length 8 cm is at a distance of 3 cm from the center of a circle. Find the radius of the circle.
Reveal Answer & Explanation
Answer: The perpendicular from the center bisects the chord: half chord $= 8/2 = 4\text{ cm}$. Distance from center $= 3\text{ cm}$. By Pythagoras: $r = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\text{ cm}$.
Radius is 5 cm.
2
In a circle with center $O$, arc $AB$ subtends an angle of $100^\circ$ at the center. Find the angle subtended by this arc at any point on the remaining circumference.
Reveal Answer & Explanation
Answer: By Theorem 9.7: Angle at circumference $= \frac{1}{2} \times \text{Angle at center} = \frac{100^\circ}{2} = 50^\circ$.
50°.
3
What is the measure of an angle inscribed in a semicircle?
Reveal Answer & Explanation
Answer: An angle in a semicircle is a right angle ($90^\circ$).
90°.
4
If $ABCD$ is a cyclic quadrilateral with $\angle A = 85^\circ$, find the measure of its opposite angle $\angle C$.
Reveal Answer & Explanation
Answer: In a cyclic quadrilateral, opposite angles are supplementary: $\angle A + \angle C = 180^\circ \implies \angle C = 180^\circ - 85^\circ = 95^\circ$.
95°.
5
Can a parallelogram inscribed in a circle be any shape other than a rectangle? Explain.
Reveal Answer & Explanation
Answer: No. In an inscribed parallelogram, opposite angles are equal ($\angle A = \angle C$) and also supplementary ($\angle A + \angle C = 180^\circ$). Therefore, $2\angle A = 180^\circ \implies \angle A = 90^\circ$. A parallelogram with a $90^\circ$ angle is a rectangle.
A cyclic parallelogram is always a rectangle.
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