Circles Class 10 ICSE — Bansal Chapter 17
Circles Class 10 ICSE is a core geometry chapter in the Bansal Mathematics book. It builds chord properties, three central theorems about angles in a circle, cyclic quadrilaterals, and results linking arcs to angles. This page walks through every section of the Class 10 ICSE chapter in the book’s own order, with formulas and worked patterns for each theorem.
Download Circles Class 10 ICSE Mathematics Chapter 17 PDF
Download the Circles Class 10 ICSE Mathematics chapter PDF from the Bansal textbook to revise offline or print for practice. It is the full chapter as printed in the book — 25 pages, 5 sections, 61 questions.
What is inside Chapter 17
25 pages · 5 sections · 14 worked examples · 61 exercise questions
| Section | Title |
|---|---|
| 17.1 | Introduction |
| 17.2 | Arc and its types |
| 17.3 | Segment and relation between arcs and segments |
| 17.4 | Cyclic Properties |
| 17.5 | Some important results |
- EXERCISE 17(A): 61 questions
17.1 Introduction
A circle is the locus of a point that moves in a plane so its distance from a fixed point stays constant. The fixed point is the centre and the constant distance is the radius. The perimeter of the circle is its circumference. These three terms appear in every theorem that follows.
Two or more circles with the same centre but different radii are concentric circles. Circles with equal radii are equal, or congruent. A circle passing through all vertices of a polygon is the circumscribed circle, whose centre is the circumcentre. A circle touching all sides of a polygon is the inscribed circle, whose centre is the incentre.
A chord is the line segment joining any two points on the circumference. A chord through the centre is the diameter, the largest chord. The perpendicular from the centre bisects a chord. Conversely, the line from the centre that bisects a chord (not a diameter) is perpendicular to it. Equal chords are equidistant from the centre, and chords equidistant from the centre are equal.
A larger chord sits closer to the centre than a smaller one. A common mistake is confusing chord length with distance from the centre. Take a chord AB of length 8 cm. The perpendicular from centre O meets AB at M, so AM = 4 cm. If OM = 3 cm, then OA^2 = OM^2 + AM^2 = 9 + 16 = 25, so OA = 5 cm. That is the radius.
One unique circle passes through three given non-collinear points. The Class 10 ICSE Maths book hub lists every chapter in order. For the official syllabus and exam format, check the CISCE website.
17.2 Arc and its types
An arc is a part of the circumference of a circle. A chord divides the circumference into two parts, and each part is an arc. The arc smaller than a semi-circle is the minor arc. The arc larger than a semi-circle is the major arc. Both arcs share the same endpoints, the chord’s endpoints.
Every chord therefore creates two arcs. When a question names an arc, check which side of the chord it lies on. That decides whether the arc is minor or major. This distinction matters because the angle at the centre on the major arc uses the reflex angle, while the minor arc uses the smaller angle.
No formula is needed for arcs in this section. The key is recognising the arc in the diagram. A diameter creates two equal arcs, each a semi-circle. Any other chord creates one minor and one major arc. Board diagrams label arcs with three letters, like arc APB. The middle letter names a point on the arc itself.
17.3 Segment and relation between arcs and segments
A segment is the part of a circle bounded by an arc and a chord. The three theorems in this section connect arcs, segments, and the centre of the circle.
Theorem 5 states that the angle an arc subtends at the centre is double the angle it subtends at any point on the remaining circumference. In symbols, \angle AOB = 2\angle ACB. The proof joins CO and produces it to D, then uses the exterior angle property of a triangle twice.
Worked pattern: if \angle AOB at the centre is 80°, then any angle at the circumference standing on the same arc, say \angle ACB, is 40°. When the arc is large, use the reflex angle at the centre: reflex \angle AOB = 2\angle APB.
Theorem 6 says angles in the same segment of a circle are equal. That is, \angle ACB = \angle ADB when C and D lie on the same arc. Example: if \angle ACB = 35°, then \angle ADB = 35°. Both stand on the same chord AB.
Theorem 7 states that the angle in a semi-circle is a right angle. If AB is a diameter, then any angle ACB standing on it equals 90°. This follows because \angle AOB = 180°, and the angle at the centre is double the angle at the circumference.
| Theorem | Statement | Formula |
|---|---|---|
| Theorem 5 | Angle at centre is double the angle at remaining circumference | \angle AOB = 2\angle ACB |
| Theorem 6 | Angles in the same segment are equal | \angle ACB = \angle ADB |
| Theorem 7 | Angle in a semi-circle is a right angle | \angle ACB = 90° |
A common error is applying the angle-at-centre rule to the wrong arc. Always confirm which arc the angle at the circumference stands on. If the angle lies on the major arc, the reflex angle at the centre doubles it, not the smaller angle.
17.4 Cyclic Properties
A quadrilateral inscribed in a circle, with all four vertices on the circumference, is a cyclic quadrilateral. Points that lie on the circumference of the same circle are concyclic points.
Theorem 8 states that the opposite angles of a cyclic quadrilateral are supplementary. For quadrilateral ABCD, \angle ABC + \angle ADC = 180° and \angle BAD + \angle BCD = 180°. The proof joins OA and OC. Arc ABC subtends \angle AOC at the centre and \angle ADC at D, so \angle AOC = 2\angle ADC. Similarly, \angle ABC equals half the reflex \angle AOC.
Adding the two gives \frac{1}{2} \times 360° = 180°.
The converse also holds: if the opposite angles of a quadrilateral are supplementary, the quadrilateral is cyclic. Also, the angle of the major segment is acute, and the angle of the minor segment is obtuse.
Theorem 9 states that the exterior angle of a cyclic quadrilateral equals the interior opposite angle. If side AB is produced to E, then \angle CBE = \angle ADC. This follows from Theorem 8 because \angle ABC + \angle CBE = 180° and \angle ABC + \angle ADC = 180°.
Worked pattern: in a cyclic quadrilateral ABCD with centre O, suppose \angle AOC = 120°. Then \angle ADC = 60°, half the angle at the centre. And \angle ABC = 120°, supplementary to \angle ADC. This single example combines Theorems 5 and 8.
Common mistakes when applying cyclic properties:
- Assuming any quadrilateral is cyclic. Only quadrilaterals with supplementary opposite angles are cyclic.
- Applying the exterior angle theorem to a non-cyclic figure.
- Confusing which angle pairs are supplementary.
Board questions in the Class 10 ICSE circles chapter often ask you to prove that a line is a diameter. The standard path is to show the angle subtended by that line is 90°. If \angle ADC = 90°, then AC is a diameter of the circle. The next chapter on tangents and intersecting chords builds on the same angle theorems.
17.5 Some important results
Section 17.5 collects results that appear repeatedly in exercises. Equal chords subtend equal angles at the centre. If AB = CD, then \angle AOB = \angle COD, and the converse holds. For a regular polygon inscribed in a circle, the central angle is \frac{360°}{n}, where n is the number of sides.
| Polygon | Sides | Central angle |
|---|---|---|
| Triangle | 3 | 120° |
| Square | 4 | 90° |
| Pentagon | 5 | 72° |
| Hexagon | 6 | 60° |
Arc lengths in a circle are proportional to the angles they subtend at the centre. If chord AB : chord CD = 7 : 5, then \angle AOB : \angle COD = 7 : 5.
Worked pattern: suppose arc AB : arc BC = 3 : 2 and \angle AOB = 96°. Then \angle BOC = \frac{2}{3} \times 96° = 64°. Since \angle BOC = 2\angle CAB, we get \angle CAB = 32°.
If two sides of a cyclic quadrilateral are parallel, the other two sides are equal. This makes the figure a cyclic trapezium, with equal non-parallel sides and equal diagonals. The proof uses co-interior angles and the supplementary opposite-angle property.
Do not mix arc ratio with chord length. The ratio applies to the angles at the centre, not to the straight-line distance across the circle. These results appear throughout the Class 10 ICSE circles exercises.
What to practise from Circles Class 10 ICSE
Exercise 17(A) contains 61 questions covering every theorem in this chapter. Start with the first ten questions to build confidence with the angle-at-centre rule. Then attempt the board-style questions marked with years such as 2003, 2005, 2006, 2007 and 2013. These mirror the CISCE exam pattern.
- Exercise 17(A) — 61 questions covering chord properties, all three angle theorems, cyclic quadrilaterals, and arc ratios.
Circles Class 10 ICSE frequently asked questions
What is the angle in a semi-circle in Class 10 ICSE circles?
The angle in a semi-circle is always 90°, a right angle. This is Theorem 7. If AB is a diameter and C is any point on the circumference, then \angle ACB = 90°.
How do you prove a quadrilateral is cyclic?
Show that its opposite angles are supplementary, meaning they sum to 180°. This is the converse of Theorem 8. If \angle ABC + \angle ADC = 180° and \angle BAD + \angle BCD = 180°, the quadrilateral is cyclic.
How do you find unknown angles using arc ratios?
Arc lengths are proportional to the angles they subtend at the centre. If arc AB : arc BC = 3 : 2 and \angle AOB = 96°, then \angle BOC = \frac{2}{3} \times 96° = 64°. Then use the angle-at-centre rule to halve it for the angle at the circumference.
How does the exterior angle theorem work for cyclic quadrilaterals?
When a side of a cyclic quadrilateral is extended, the exterior angle equals the interior opposite angle. If side AB is produced to E, then \angle CBE = \angle ADC. This is Theorem 9, a direct consequence of Theorem 8.
Reference: ICSE Class 10 Mathematics textbook material, chapter 17: Circles — CISCE-aligned.
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