The Circle Reference Content Class 6 ICSE — Maths Chapter 21
The Circle Reference Content Class 6 ICSE is Chapter 21 of the Class 6 Mathematics book. It teaches the parts of a circle: radius, diameter, chord, secant, tangent, arc, sector and segment. It also covers the interior and exterior of a circle. The chapter ends with constructing the circumcircle and incircle of a triangle. Every term gets a clear definition and a supporting figure.
Download The Circle Reference Content Class 6 ICSE Mathematics Chapter 21 PDF
Download the The Circle Reference Content Class 6 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 5 pages, 11 sections, 18 questions.
What is inside Chapter 21
5 pages · 11 sections · 18 exercise questions
| Section | Title |
|---|---|
| 21.1 | CIRCLE |
| 21.2 | RADIUS |
| 21.3 | DIAMETER |
| 21.4 | CHORD |
| 21.5 | SECANT AND TANGENT |
| 21.6 | ARC |
| 21.7 | SECTOR |
| 21.8 | SEGMENT |
| 21.9 | INTERIOR AND EXTERIOR OF A CIRCLE |
| 21.10 | CIRCUMCIRCLE OF A TRIANGLE |
| 21.11 | IN-CIRCLE OF A TRIANGLE |
- EXERCISE 21(A): 8 questions
- EXERCISE 21(B): 5 questions
- Revision Exercise (Chapter 21): 5 questions
21.1 CIRCLE
This Class 6 ICSE chapter opens by defining a circle. A circle is a closed curve such that every point on its boundary is at the same distance from a fixed point inside it. That fixed point is called the centre.
The book also gives a second definition. A circle is the path traced by a moving point that always stays at a fixed distance from a fixed point. The path traced is called the circumference. The fixed distance is the radius.
Why two definitions? The first helps you recognise a circle as a shape. The second explains how the shape is produced. Both describe the same figure, but each fits a different situation.
Think of a compass. The sharp point is the fixed centre. The pencil end is the moving point. Keeping the gap constant traces a perfect circle. Every point on that path is the same distance from the centre.
The three terms — centre, circumference and radius — appear throughout the chapter. Master them here, because every later section builds on them.
21.2 RADIUS
The radius of a circle is the length of the line segment joining the centre to any point on its circumference.
In the figure, OP joins the centre O to point P on the circumference. Therefore OP is the radius of that circle.
The radius is a length, not a line. It is measured in units such as centimetres or metres. Every radius of the same circle has the same length. That is why all points on the circumference are equidistant from the centre.
This section is short, but it matters. The radius feeds directly into the diameter relation in section 21.3. It also appears in the position rule that follows, where distances are compared with the radius.
21.3 DIAMETER
A diameter is a straight line passing through the centre of a circle, with both ends on the circumference. In the figure, AB is a diameter.
The key relation connects diameter and radius:
\text{Diameter} = 2 \times \text{Radius}
\text{Radius} = \frac{\text{Diameter}}{2}
A diameter always divides the circle into two equal parts. Each part is called a semi-circle.
The section then states an important position rule. Let a circle have centre O and let point P lie in the same plane. Compare the distance OP with the radius:
- If OP is greater than the radius, P lies outside the circle.
- If OP equals the radius, P lies on the circumference.
- If OP is less than the radius, P lies inside the circle.
Worked pattern: take a circle of radius 5 cm. If OP = 7 cm, then 7 \gt 5, so P lies outside. If OP = 5 cm, then 5 = 5, so P lies on the circumference. If OP = 3 cm, then 3 \lt 5, so P lies inside.
The comparison is all you need. You never have to measure the circle itself. The position rule is a favourite in the Class 6 ICSE syllabus.
21.4 CHORD
A chord is a straight line joining any two points on the circumference of a circle.
In the figure, PQ joins points P and Q on the circumference, so PQ is a chord. The diameter AB is also a chord, because both its ends lie on the circumference.
The section makes two points. First, a chord passing through the centre is a diameter. Second, the diameter is the longest chord of the circle.
A common mistake: students treat chord and diameter as different shapes. In fact, every diameter is a chord, but only the chord that passes through the centre is a diameter. A chord drawn away from the centre is not a diameter.
Use the centre as your test. Does the chord pass through the centre? If yes, it is a diameter. If no, it is just a chord.
21.5 SECANT AND TANGENT
A secant is a straight line that intersects a circle at two points. A tangent is a straight line that touches the circle at one point only.
In the figure, line AB cuts the circle at points M and N, so AB is a secant. Line CD touches the circle at point P only, so CD is a tangent. Point P is the point of contact.
| Line | Meets the circle at | Passes through the circle? |
|---|---|---|
| Secant | Two points | Yes — cuts across |
| Tangent | One point | No — touches only |
Why does this distinction matter? A secant enters the circle and leaves it, so it has two intersection points. A tangent just grazes the circle, so it has exactly one.
In later construction work, a tangent is drawn at right angles to the radius at the point of contact. This chapter defines the terms; the perpendicular property appears in later classes.
21.6 ARC
An arc is a part of the circumference of a circle.
Take two points A and B on the circumference. Joining them divides the circumference into two arcs. The shorter arc is the minor arc. The longer arc is the major arc.
In the figure, arc APB is the minor arc and arc AQB is the major arc. The names depend on length, not on position. Any short part is minor, and the remaining longer part is major.
There is one special case. If A and B are opposite ends of a diameter, the two arcs are equal in length. Each is then a semi-circle.
21.7 SECTOR
A sector is the part of a circle enclosed by two radii and an arc.
In the figure, the radii OA and OB together with the arc APB enclose the sector OAPB. The two radii divide the circle into two sectors. The shaded minor sector is OAPB. The unshaded major sector is OAQB.
The section adds a note. A quadrant is a sector enclosed by two mutually perpendicular radii. In other words, when the angle between the two radii is 90 degrees, the sector is a quadrant.
A quadrant is one quarter of the circle. You will meet it in drawing questions, where you shade the minor sector between two radii set at a right angle.
Sector and segment sound similar, so students mix them up. Remember: a sector uses two radii and an arc. A segment uses a chord and an arc. The chord is straight; the radii meet at the centre.
21.8 SEGMENT
A segment is a part of a circle enclosed by a chord and an arc.
Every chord divides the circle into two parts. The smaller part is the minor segment. The larger part is the major segment.
The section gives two facts. First, the major segment contains the centre of the circle, while the minor segment does not. Second, a diameter divides the circle into two equal parts, each called a semi-circle.
The table below is a handy reference for this ICSE chapter:
| Term | Enclosed by | Minor part | Major part |
|---|---|---|---|
| Arc | — | Shorter arc APB | Longer arc AQB |
| Sector | Two radii + an arc | Minor sector OAPB | Major sector OAQB |
| Segment | A chord + an arc | Minor segment (no centre) | Major segment (contains centre) |
The common mix-up: a sector has a pointed vertex at the centre, like a slice of pie. A segment has a straight edge (the chord) and a curved edge (the arc), like a crescent. Draw both to see the difference.
If the chord is a diameter, the two segments are equal semi-circles.
21.9 INTERIOR AND EXTERIOR OF A CIRCLE
The interior of a circle is the set of points in the plane that lie inside the circle. The exterior is the set of points that lie outside it. Together with the circumference, they divide the plane into three regions.
This section restates the position rule from section 21.3 in new words. A point whose distance from the centre is less than the radius lies in the interior. A point whose distance is greater than the radius lies in the exterior. A point whose distance equals the radius lies on the circumference.
The circumference itself is the boundary. Points on it are neither interior nor exterior.
In the figure, points O, A, B and C lie inside. Points D, E and F lie outside. The interior is shaded. The same rule answers the revision questions that ask where a point lies.
21.10 CIRCUMCIRCLE OF A TRIANGLE
A circumcircle is a circle that passes through all three vertices of a triangle.
To construct the circumcircle of a given triangle ABC:
- Draw the perpendicular bisectors of any two sides of the triangle.
- Let the bisectors of sides BC and AC meet at point O.
- Draw a circle with centre O and radius OA, OB or OC.
This circle passes through all three vertices, A, B and C.
The point O, where the perpendicular bisectors meet, is the circumcentre. The radius of the circle is the circumradius. It satisfies OA = OB = OC.
Worked pattern: take a triangle with sides AB = 5 cm, BC = 6 cm and AC = 7 cm. Draw the perpendicular bisector of BC. Draw the perpendicular bisector of AC. They meet at O. With centre O and radius OA, draw the circle. It passes through A, B and C. Measure OA to find the circumradius.
Why perpendicular bisectors? Every point on a perpendicular bisector is equidistant from the two ends of the segment it bisects. Where two bisectors meet, the point is equidistant from all three vertices. That is exactly what a circle through three points needs. For more triangle work, visit the Class 6 ICSE Maths book hub.
21.11 IN-CIRCLE OF A TRIANGLE
An in-circle is a circle drawn inside a triangle such that it touches all three sides.
To construct the in-circle of a given triangle ABC:
- Draw the bisectors of any two angles of the triangle.
- Let these bisectors meet at point I.
- From I, drop a perpendicular IP to side BC.
- With centre I and radius IP, draw a circle.
This circle touches all three sides of the triangle.
The point I, where the angle bisectors meet, is the incentre. The length IP is the inradius.
Worked pattern: take a triangle with sides AB = 4.5 cm, BC = 5.5 cm and AC = 6.5 cm. Draw the bisectors of angle A and angle B. They meet at I. Drop a perpendicular from I to BC, meeting at P. With centre I and radius IP, draw the circle. It touches AB, BC and AC at one point each. Construction questions like this appear across the Class 6 ICSE syllabus.
Why angle bisectors? Every point on an angle bisector is equidistant from the two sides of that angle. Where two bisectors meet, the point is equidistant from all three sides. So one circle centred at I can touch all of them.
What to practise from The Circle Reference Content Class 6 ICSE
EXERCISE 21(A) has 8 questions. It covers definitions, fill-in-the-blanks and drawing circles with given radii and diameters.
EXERCISE 21(B) has 5 questions. It focuses on constructing the circumcircle and incircle of a triangle.
Revision Exercise (Chapter 21) has 5 questions. It tests the position rule, the radius-diameter relation, chords, sectors and true-or-false statements.
Work through the exercises in order. See the full list of Class 6 ICSE books for other subjects. You can verify the syllabus on the official CISCE website.
The Circle Reference Content Class 6 ICSE frequently asked questions
What is the circle reference content class 6 ICSE about?
It is Chapter 21 of the Class 6 ICSE Mathematics book. It defines the parts of a circle and explains how to construct circumcircles and incircles.
How is a diameter different from a chord?
A diameter passes through the centre; a chord does not have to. Every diameter is a chord, but only the chord through the centre is a diameter. The diameter is also the longest chord.
What is the difference between a secant and a tangent?
A secant intersects the circle at two points. A tangent touches it at one point only. That single point is the point of contact.
How do I construct the circumcircle and incircle of a triangle?
For the circumcircle, draw the perpendicular bisectors of two sides; their meeting point is the circumcentre. For the incircle, draw the angle bisectors of two angles; their meeting point is the incentre.
What is a quadrant in a circle?
A quadrant is a sector enclosed by two mutually perpendicular radii. It is one quarter of the circle.
Reference: ICSE Class 6 Mathematics textbook material, chapter 21: The Circle Reference Content — CISCE-aligned.