⭕ Maths Class 9 NCERT Lesson 5
Circles
Special Edition for Easy Learning
What is inside?
Section 1 — Classroom Notes
Section 2 — Illustrations
Section 3 — Worked Examples
Section 4 — Cheat Sheet
Section 5 — Handout
Section 6 — Revision Flashcards
Section 7 — Mind Map
📝 Section 1 — Classroom Notes
🌎 Real-World Hook: Look at a bicycle wheel, a clock face, a coin, or the pupils of your eyes — they are all circles! A circle is the most perfect shape in nature. Planets orbit in paths close to circles. A dartboard has rings that are circles. Even a pizza is circular! Whenever engineers design wheels, pipes, or domes, they use the properties of circles. In this lesson, you will discover the beautiful mathematical rules that govern this perfect shape.
Topic 1: What is a Circle?
A circle is a round shape.
It has a centre point.
All points on it are the same distance from centre.
That distance is called the radius.
A line across the full circle = diameter.
Diameter = 2 × radius.
A line joining two points on circle = chord.
The diameter is the longest chord.
Topic 2: Symmetry of a Circle
A circle has perfect symmetry.
Rotate it by any angle. It looks the same!
Every diameter is a line of symmetry.
So a circle has infinite lines of symmetry.
No other shape has this much symmetry.
Topic 3: Circles Through Points
Through 1 point: infinite circles possible.
Through 2 points: still infinite circles.
Through 3 points (not in a line): only ONE circle.
This one circle is called the circumcircle.
3 points in a straight line: no circle possible.
Topic 4: Circumcircle of a Triangle
Draw a triangle. A circle passes through all 3 corners.
That circle is the circumcircle.
Its centre is the circumcentre.
To find it: draw the perpendicular bisector of each side.
All three bisectors meet at ONE point. That is the circumcentre.
Acute triangle: circumcentre is inside.
Obtuse triangle: circumcentre is outside.
Right triangle: circumcentre is at midpoint of hypotenuse.
Topic 5: Equal Chords and Angles
Two chords of same length make same angle at centre.
This is because the triangles formed are congruent (SSS).
Also works the other way:
Two chords that make same angle at centre have same length.
Topic 6: Centre to Chord is 90°
Draw a line from centre to midpoint of a chord.
This line is always at 90° to the chord.
Also works the other way:
Drop a perpendicular from centre to a chord.
It always hits the midpoint of the chord.
Topic 7: Equal Chords = Same Distance
Chords of same length are at same distance from centre.
Chords at same distance have same length.
Longer chord = closer to the centre.
Shortest distance is 0 (the diameter goes through centre).
The diameter is the longest chord.
Topic 8: Major Arc and Minor Arc
Two points on a circle split it into two arcs.
The bigger part = major arc.
The smaller part = minor arc.
If both arcs are equal, each is a semicircle.
Topic 9: Angle at Centre = 2 × Angle on Circle
An arc makes an angle at the centre.
The same arc makes an angle at a point on the circle.
The angle at the centre is double the angle on the circle.
Centre angle = 2 × circle angle.
Special case: diameter makes 90° at any point on circle.
All angles in the same arc are equal.
Angle at centre = 2 × Angle on circle
Topic 10: Cyclic Quadrilateral
4 points on one circle = concyclic points.
A quadrilateral with all 4 corners on a circle = cyclic quadrilateral.
Opposite angles always add up to 180°.
If opposite angles add to 180°, it IS cyclic.
Opposite angles of cyclic quadrilateral = 180°
🎨 Section 2 — Illustrations
Picture 1: Parts of a Circle
A circle showing centre, radius, diameter, chord, and arcs.
Picture 2: Circumcircle of a Triangle
One unique circle passes through all 3 corners of a triangle.
Picture 3: Centre to Chord = 90°
CM is perpendicular to chord AB. It also bisects the chord.
Picture 4: Angle at Centre = 2 × Angle on Circle
Arc AB makes angle 2x at centre O and angle x at point D on the circle.
Picture 5: Cyclic Quadrilateral
In a cyclic quadrilateral, opposite angles always add up to 180°.
📐 Section 3 — Worked Examples
Example 1: Find the Chord Length
A circle has radius 13 cm.
The chord is 5 cm away from the centre.
Find the length of the chord.
1 Draw the picture. Centre C, chord AB, midpoint M.
2 CM = 5 cm (distance to chord).
3 CA = 13 cm (radius).
4 Triangle CMA has a 90° angle at M.
5 Use Pythagoras: AM² = CA² − CM²
6 AM² = 13² − 5² = 169 − 25 = 144
7 AM = √144 = 12 cm
8 AB = 2 × AM = 2 × 12 = 24 cm
Chord length = 24 cm
Formula used: AM² + CM² = radius²
Example 2: Find the Angle on the Circle
An arc makes a 70° angle at the centre.
What angle does it make at a point on the circle?
1 The rule: centre angle = 2 × circle angle.
2 Centre angle = 70°
3 Circle angle = 70° ÷ 2 = 35°
Angle on circle = 35°
Rule: Centre angle = 2 × circle angle
Example 3: Is it a Cyclic Quadrilateral?
A quadrilateral has angles: 80°, 100°, 100°, 80°.
Is it cyclic?
1 Check: do opposite angles add to 180°?
2 Pair 1: 80° + 100° = 180° ✔
3 Pair 2: 100° + 80° = 180° ✔
4 Both pairs add to 180°. Yes, it is cyclic!
Yes, it is a cyclic quadrilateral.
Rule: Opposite angles must add to 180°
📋 Section 4 — Cheat Sheet
Word / Rule
What It Means
Circle
All points same distance from centre
Radius
Distance from centre to circle
Diameter
Longest chord. Passes through centre. d = 2r
Chord
A line joining two points on the circle
Arc
A curved part of the circle
Circumcircle
Circle through all 3 corners of a triangle
Circumcentre
Centre of circumcircle. Where bisectors meet
Equal chords
Make equal angles at centre. Same distance from centre
Centre to chord
Line from centre to midpoint is 90° to chord
Centre angle rule
Centre angle = 2 × angle on circle
Angle in semicircle
Always 90°
Cyclic quadrilateral
All 4 corners on one circle. Opposite angles = 180°
✏️ Section 5 — Handout
A. Fill in the Blanks
1. The distance from centre to circle is ________.
2. Diameter = 2 × ________.
3. The longest chord of a circle is the ________.
4. Through 3 non-collinear points, there is ________ circle.
5. Centre angle is ________ times the circle angle.
B. True or False
1. A circle has only 4 lines of symmetry. ________
2. Equal chords make equal angles at centre. ________
3. Angle in a semicircle is 90°. ________
4. The diameter is the shortest chord. ________
C. Solve
1. Radius = 10 cm. Distance to chord = 6 cm.
Find the chord length.
Show your work here
2. Centre angle = 120°. Find the angle on the circle.
Show your work here
D. Match the Columns
1. Circumcentre — (a) Longest chord
2. Diameter — (b) Opposite angles = 180°
3. Cyclic quadrilateral — (c) Where bisectors meet
4. Radius — (d) Centre to circle distance
E. Draw and Label
Draw a circle. Mark and label:
Centre, Radius, Diameter, Chord, Arc.
Draw your circle here
✅ Detailed Solutions — Handout Practice
✓ A. Fill in the Blanks — Answers
1 The distance from centre to circle is radius.
2 Diameter = 2 × radius.
3 The longest chord of a circle is the diameter.
4 Through 3 non-collinear points, there is exactly one circle.
5 Centre angle is twice (2 times) the circle angle (inscribed angle theorem).