What is the angle in a semicircle?

The angle subtended by a diameter at any point on the circle is always 90 degrees.

What is a cyclic quadrilateral?

A quadrilateral whose all four vertices lie on a circle is called a cyclic quadrilateral.

What is the relation between angle at centre and angle at the circle?

The angle subtended by an arc at the centre is twice the angle subtended by it at any point on the remaining part of the circle.

What happens when a perpendicular is drawn from the centre to a chord?

The perpendicular drawn from the centre of a circle to a chord bisects the chord.

Class 9 Maths Chapter 9 Circles | NCERT Notes, Theorems & MCQs

Why Study Circles?

Used in wheels, clocks, coins, gears, fans
Foundation for Areas of Circles (Class 10)
Important geometry theorem-based chapter
Very high weightage in proof + MCQs

1. Anatomy of a Circle

A circle is the collection of all points in a plane, which are at a fixed distance (radius) from a fixed point (center).

Click parts of the circle:

Select a part...

Key Definitions (Write Exactly Like This)

Circle: Set of all points in a plane at a fixed distance from a fixed point.
Radius: Distance from centre to any point on the circle.
Diameter: Longest chord of the circle (2 × radius).
Chord: Line joining any two points on the circle.
Arc: A part of the circumference.

2. Chords and the Center

Theorem 9.1: Equal Chords

Equal chords of a circle subtend equal angles at the center.

Theorem 9.3: Perpendicular from Center

The perpendicular from the center of a circle to a chord bisects the chord.

Theorem 9.6: Equidistant Chords

Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).

Perpendicular Drop

Click "Drop" to see the theorem in action.

Circles – Theorem Map 🧠

Equal chords subtend equal angles at the centre
Equal chords are equidistant from the centre
Perpendicular from centre bisects the chord
Angle at centre = 2 × angle at circumference
Angles in same segment are equal
Angle in semicircle is 90°
Opposite angles of cyclic quadrilateral are supplementary

3. Angles Subtended by an Arc

Theorem 9.8: 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.

$$ \angle AOB = 2 \times \angle APB $$

Angle in the Same Segment

Theorem: Angles in the same segment of a circle are equal.

Exam Tip:
If two angles subtend the same chord from the same side, they are equal.

Theorem 9.10 (Concyclic Points):
If a line segment joining two points subtends equal angles at two other points on the same side, the four points lie on a circle (they are concyclic).

Angle in a Semi-Circle

Theorem: The angle subtended by a diameter at any point on the circle is 90°.

If AB is diameter → ∠APB = 90°

4. Cyclic Quadrilaterals

A quadrilateral is called cyclic if all its four vertices lie on a circle.

Property: The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

Cyclic Quad Solver

If $ \angle A $ is given, find opposite $ \angle C $.

How to Write Proofs (CBSE Style)

Write Given
Write To Prove
Join required lines (construction)
Apply known theorem
Conclude with statement

Always write theorem name before using it ✔

Common Mistakes to Avoid ❌

❌ Angle at centre is NOT equal to angle at circle
❌ All quadrilaterals are NOT cyclic
❌ Radius perpendicular to chord only at midpoint
❌ Diameter is the ONLY longest chord

🏆 NCERT Exemplar (Advanced)

Targeting full marks? Solve these higher-level problems from NCERT Exemplar.

9.1 MCQs 9.2 Short Ans 9.3 Short Ans 9.4 Long Ans

Assertion–Reason Practice

Assertion (A): The angle in a semicircle is a right angle.

Reason (R): The angle subtended at the centre is twice the angle at the circle.

✔ A and R are true, and R explains A

Circles in Real Life 🌍

Wheels rotate around a fixed centre
Clock angles follow circle theorems
Coins & plates are perfect circles
Satellite paths use circular geometry

One-Page Revision

Diameter = 2 × Radius
Angle at centre = 2 × angle at circumference
Angle in semicircle = 90°
Same segment → equal angles
Cyclic quad → opposite angles = 180°

🎯 Exam Smart Zone

Common board-style questions. Click to reveal simplified solutions.

Q1: Two chords AB and CD of a circle are equal. Prove that their corresponding arcs are congruent.

Given: AB = CD (equal chords)

To Prove: arc(AB) ≅ arc(CD)

Proof: Equal chords subtend equal angles at the centre.

∠AOB = ∠COD (equal chords → equal central angles)

∴ arc(AB) = arc(CD) (arcs subtending equal central angles are congruent)

Q2: In a cyclic quadrilateral ABCD, if ∠A = 70°, find ∠C.

Property: Opposite angles of a cyclic quadrilateral are supplementary.

∠A + ∠C = 180°

70° + ∠C = 180°

∴ ∠C = 110°

Q3: A chord of length 8 cm is at a distance of 3 cm from the centre. Find the radius.

Key: Perpendicular from centre bisects the chord.

Half chord = 8/2 = 4 cm, Distance = 3 cm

By Pythagoras: r² = 4² + 3² = 16 + 9 = 25

∴ r = 5 cm

Q4: Prove that the angle subtended by an arc at the centre is double the angle subtended at any point on the remaining part.

Theorem: Central Angle Theorem

Let arc PQ subtend ∠POQ at centre and ∠PAQ at point A on circle.

Join AO and extend to R.

In △OAP: OA = OP (radii) → ∠OAP = ∠OPA (isosceles triangle)

∠POR = 2∠OAP (exterior angle theorem)

Similarly, ∠QOR = 2∠OAQ

∴ ∠POQ = 2∠PAQ

Class 9 Maths Chapter 9: Circles