Circles Class 9 ICSE — Maths Chapter 17
Circles class 9 ICSE maths (Chapter 17 of your Class 9 Mathematics book) builds the geometry of the circle from its definition up to the theorems that govern chords and arcs. The chapter starts with circles you already know — a wheel, a clock’s minute hand — then fixes the exact terms: centre, radius, circumference, chord and diameter. It ends with the chord and arc properties that power most board proofs.
Download Circle Class 9 ICSE Mathematics Chapter 17 PDF
Download the Circle Class 9 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 19 pages, 5 sections, 28 questions.
What is inside Chapter 17
19 pages · 5 sections · 11 worked examples · 28 exercise questions
| Section | Title |
|---|---|
| 17.1 | INTRODUCTION |
| 17.2 | CIRCLE |
| 17.3 | MORE ABOUT CIRCLE |
| 17.4 | ARC, SEGMENT AND SECTOR |
| 17.5 | CHORD PROPERTIES |
- EXERCISE 17(A): 10 questions
- EXERCISE 17(C): 8 questions
- EXERCISE 17(D): 10 questions
17.1 INTRODUCTION
Section 17.1 keeps things concrete. The wheel of a bicycle or car is the first picture the chapter shows you. The path traced by the tip of a wall clock’s minute hand is the second. Both are circles you have seen every day.
The point is not to define anything yet. It is to get you to see that the same closed shape keeps appearing. That makes the formal definition in 17.2 easier to hold on to.
17.2 CIRCLE
A circle is the closed curve formed by all those points in a plane which are at the same fixed distance from a fixed point in the same plane. Another way to say it: a circle is the locus of a point that moves in a plane so that its distance from a fixed point stays constant.
Those two phrases say the same thing. The fixed point is the centre, and the fixed distance is the radius. The perimeter of the circle is its circumference.
A chord is a line segment joining any two points on the circumference. A chord that passes through the centre is a diameter, and it is the largest chord of the circle. The diameter is twice the radius:
d = 2r
Why is the diameter the largest chord? Any other chord is a shorter straight line inside the circle. The diameter spans the full width through the centre, so no chord can beat it.
Worked pattern: a circle has a diameter of 24 cm. The radius is half of that, so r = \frac{24}{2} = 12\ \text{cm}. You will use this relation constantly, because most chord questions end with the radius.
17.3 MORE ABOUT CIRCLE
Any point in the plane of a circle falls into one of three classes. The class depends on how its distance from the centre compares with the radius r.
| Position of point | Condition |
|---|---|
| Exterior point | distance from centre \gt r |
| Point on the circumference | distance from centre = r |
| Interior point | distance from centre \lt r |
The section then defines five special circles. Concentric circles share the same centre but have different radii. Equal (congruent) circles have equal radii.
A circumscribed circle passes through all the vertices of a polygon. Its centre is the circumcentre, and the polygon is called an inscribed polygon. An inscribed circle touches all the sides of a polygon. Its centre is the incentre, and the polygon is called a circumscribed polygon.
Why two names for the polygon? The names flip depending on whether the circle goes around the polygon or sits inside it. The circle is circumscribed around the polygon, or inscribed inside it.
17.4 ARC, SEGMENT AND SECTOR
An arc is a part of the circumference. A chord divides the circumference into two arcs. If the arcs are unequal, the smaller one is the minor arc and the bigger one is the major arc. If they are equal, each is a semi-circle.
A minor arc is always smaller than a semi-circle, and a major arc is always bigger. Unless the question says otherwise, the word “arc” means the minor arc.
A segment is the region bounded by an arc and a chord. The smaller region is the minor segment and the bigger one is the major segment. The centre of the circle lies in the major segment.
A sector is the region bounded by an arc and the two radii joining the centre to the ends of the arc. The minor arc gives the minor sector and the major arc gives the major sector.
| Part | Minor | Major |
|---|---|---|
| Arc | smaller part of the circumference | bigger part of the circumference |
| Segment | region smaller than a semi-circle | region bigger than a semi-circle |
| Sector | region of the minor arc | region of the major arc |
Why does the centre sit in the major segment? The major segment is the larger region, and the centre lies nearer the bigger side of the cut. In a semi-circle the two segments are equal, and so are the two sectors.
17.5 CHORD PROPERTIES
This section is the core of the circle chapter class 9 ICSE. It states seven theorems, Theorems 22 to 28, each with a proof built on congruent triangles.
Theorem 22: a straight line drawn from the centre to bisect a chord, which is not a diameter, is at right angles to the chord. The proof joins the centre to the ends of the chord and shows the two triangles congruent by S.S.S. The two angles at the chord are then equal, and they add to 180°, so each must be 90°.
Theorem 23 is the converse: the perpendicular drawn from the centre to a chord bisects the chord. This one uses R.H.S. congruence, and it is the step you apply first in most chord-length problems.
Theorem 24: equal chords of a circle are equidistant from the centre. Theorem 25 is its converse: chords equidistant from the centre are equal. Together they let you swap equal chords for equal distances in either direction.
Theorem 26: one and only one circle passes through three points not in a straight line. The perpendicular bisectors of two chords meet at the unique centre, which is equidistant from all three points.
Theorem 27: if two arcs of the same circle subtend equal angles at the centre, the arcs are equal. Theorem 28 is its converse: equal arcs subtend equal angles at the centre. Equal arcs also cut equal chords, and equal chords cut equal arcs.
Two extra rules complete the picture. A bigger chord sits closer to the centre than a smaller one. And if an n-sided regular polygon is inscribed in a circle, each side subtends an angle of 360^\circ / n at the centre.
Worked pattern: a chord of length 30 cm lies at a distance 8 cm from the centre. Find the radius.
Step 1: The perpendicular from the centre bisects the chord. A 30 cm chord splits into two halves of 15 cm.
Step 2: The radius, the half-chord and the distance from the centre form a right triangle.
r^2 = 8^2 + 15^2 = 64 + 225 = 289
r = \sqrt{289} = 17
Final answer: the radius is 17\ \text{cm}.
Common mistakes to avoid:
- Forgetting to halve the chord. Pythagoras uses the half-length, never the full chord.
- Applying Theorem 22 to a line that does not pass through the centre. The line must come from the centre.
- Reading the minor arc as the major arc when the figure is not drawn to scale.
- Mixing up a segment (arc and chord) with a sector (arc and two radii).
- Assuming a longer chord is further from the centre. It is actually closer.
Mnemonic for the seven theorems: “Right at the cut, half at the drop, equal far for equal chords, one circle for three points, equal angles for equal arcs.” That covers bisect-makes-right, perpendicular-halves, equidistance, the three-point circle, and the arc-angle pair.
What to practise from Circles Class 9 ICSE
The exercises in this chapter line up with the theorems in a clear way. Each block drills a particular skill, so practise them in order.
- EXERCISE 17(A) — 10 questions on chord length and distance from the centre, drawing on Theorems 22 to 25.
- EXERCISE 17(C) — 8 questions on arcs and angles at the centre, drawing on Theorems 27 and 28 and the rule 360^\circ / n for regular polygons.
- EXERCISE 17(D) — 10 mixed questions including proofs, pulling together all the theorems of the chapter.
This ICSE chapter sits inside the wider Class 9 Mathematics course; see the Class 9 Mathematics hub for the chapters around it. The Class 9 books list holds the other subjects, and the official CISCE syllabus states the full board requirements.
Circles Class 9 ICSE frequently asked questions
What is a chord in circles class 9 ICSE?
A chord is a line segment joining any two points on the circumference of a circle. A chord that passes through the centre is a diameter, the largest chord of the circle.
How do I find the distance of a chord from the centre in circles class 9 ICSE?
Draw the perpendicular from the centre to the chord; it bisects the chord. Halve the chord, then apply Pythagoras to the right triangle formed by the radius, the half-chord and the distance from the centre.
Which theorems on circles class 9 ICSE do I need to prove?
Theorems 22 to 28. That means the chord theorems, their converses, the three-point circle theorem, and the arc-angle theorems. Learn each statement with its proof method — S.S.S., R.H.S., S.A.S. or A.S.A.
What is the difference between a segment and a sector in circles class 9 ICSE?
A segment is the region bounded by an arc and a chord. A sector is the region bounded by an arc and the two radii joining the centre to the ends of the arc.
Reference: ICSE Class 9 Mathematics textbook material, chapter 17: Circle — CISCE-aligned.