Chapter Hub
I'm Up and Down, and Round and Round
This chapter builds the idea of a circle from its centre, radius, diameter, chords, arcs and angles. For CBSE-aligned preparation, you should connect each result with a clear diagram because most circle questions become simple after marking the centre, equal radii, chord, and required angles. The main exam value of this chapter is theorem use: perpendicular from centre to chord, equal chords, angle at centre and circumference, angles in the same segment, and cyclic quadrilateral properties. A good answer usually needs a labelled figure, the correct theorem statement, and one or two logical steps.
Difficulty
Medium
Study time
64-80 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 64 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
Start with one of the buckets below, then open the full map when you want the complete concept roadmap.
Circle Basics
Covers Definition of a Circle and Symmetries of a Circle.
Open mapChord Ideas
Covers Chord and Its Properties and Three Points Determine a Circle.
Open mapAngle Theorems
Covers Angle Subtended by Arc/Chord at Centre vs at Circumference and Angles in Same Segment.
Open mapCyclic Results
Covers Cyclic Quadrilateral and Concyclic Points.
Open mapKey Concepts
Concepts grouped the way the chapter is taught — open the group that matches what you want to revise.
Circle Basics
Covers Definition of a Circle and Symmetries of a Circle.
Definition of a Circle
A circle is the set of all points in a plane that are at the same distance from a fixed point called the centre. This fixed distance is called the radius.
Symmetries of a Circle
A circle has reflection symmetry about every diameter and rotational symmetry about its centre for any angle of rotation.
Chord Ideas
Covers Chord and Its Properties and Three Points Determine a Circle.
Chord and Its Properties
A chord is a line segment joining two points on a circle. The perpendicular from the centre of a circle to a chord bisects the chord, and equal chords are equidistant from the centre.
Three Points Determine a Circle
One and only one circle can pass through three non-collinear points. Its centre is found at the intersection of perpendicular bisectors of two joining segments.
Angle Theorems
Covers Angle Subtended by Arc/Chord at Centre vs at Circumference and Angles in Same Segment.
Angle Subtended by Arc/Chord at Centre vs at Circumference
The angle subtended by an arc or chord at the centre of a circle is twice the angle subtended by it at any point on the remaining part of the circle.
Angles in Same Segment
Angles in the same segment of a circle are equal. These angles stand on the same chord and have their vertices on the same arc segment.
Cyclic Results
Covers Cyclic Quadrilateral and Concyclic Points.
Cyclic Quadrilateral
A cyclic quadrilateral is a quadrilateral whose four vertices lie on one circle. In a cyclic quadrilateral, each pair of opposite angles is supplementary.
Concyclic Points
Points are called concyclic if they lie on the same circle. Four points can often be tested for concyclicity by checking whether a pair of opposite angles is supplementary.
Exam Intelligence
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
Common Traps
- Calling every chord a diameter; a diameter must pass through the centre.
- Using chord bisection without checking the perpendicular from the centre.
- Forgetting the non-collinear condition for a unique circle through three points.
- Reversing the centre-circumference angle relation.
- Applying same segment angle equality when vertices are on different segments.
- Adding adjacent angles instead of opposite angles in cyclic quadrilateral questions.
How CBSE Asks This Chapter
- Definition of a Circle
Questions may ask students to identify radius, diameter, chord, centre, or use diameter = 2r in a figure-based problem.
- Symmetries of a Circle
Students may be asked to identify lines of symmetry, rotational symmetry, or explain why a given line is not a symmetry line.
- Chord and Its Properties
Questions usually give a chord length and perpendicular distance, then ask for half-chord, full chord, or proof of equality of two chords.
- Three Points Determine a Circle
Exam questions may ask for construction of circumcircle of a triangle or proof that a circle through three non-collinear points is unique.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Remember
Radius: distance from centre to any point on the circle.Trap to avoid
Students often call the full circular region a circle.
Remember
Reflection symmetry: every diameter is an axis of symmetry.Trap to avoid
Students sometimes think only horizontal and vertical diameters are lines of symmetry.
Remember
If OM perpendicular AB, then AM = MB.Trap to avoid
Students often assume any line from the centre to a chord bisects it.
Three Points Determine a Circle
Remember
Unique circle condition: three points must be non-collinear.Trap to avoid
Students forget the condition non-collinear.
Angle Subtended by Arc/Chord at Centre vs at Circumference
Remember
For same chord or arc AB, angle at centre = 2 x angle at circumference.Trap to avoid
Students often reverse the relation and write the angle at the circumference as twice the central angle.
Remember
If P and Q are on the same segment standing on chord AB, then angle APB = angle AQB.Trap to avoid
Students apply the theorem even when the two points are on opposite segments.
Remember
For cyclic quadrilateral ABCD: angle A + angle C = 180 degrees and angle B + angle D = 180 degrees.Trap to avoid
Students sometimes add adjacent angles instead of opposite angles.
Remember
Concyclic test for four points: if one pair of opposite angles in the quadrilateral is supplementary, the four points lie on one circle.Trap to avoid
Students may say any four points are concyclic.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Definition of a Circle | Radius: distance from centre to any point on the circle. | Students often call the full circular region a circle. |
| Symmetries of a Circle | Reflection symmetry: every diameter is an axis of symmetry. | Students sometimes think only horizontal and vertical diameters are lines of symmetry. |
| Chord and Its Properties | If OM perpendicular AB, then AM = MB. | Students often assume any line from the centre to a chord bisects it. |
| Three Points Determine a Circle | Unique circle condition: three points must be non-collinear. | Students forget the condition non-collinear. |
| Angle Subtended by Arc/Chord at Centre vs at Circumference | For same chord or arc AB, angle at centre = 2 x angle at circumference. | Students often reverse the relation and write the angle at the circumference as twice the central angle. |
| Angles in Same Segment | If P and Q are on the same segment standing on chord AB, then angle APB = angle AQB. | Students apply the theorem even when the two points are on opposite segments. |
| Cyclic Quadrilateral | For cyclic quadrilateral ABCD: angle A + angle C = 180 degrees and angle B + angle D = 180 degrees. | Students sometimes add adjacent angles instead of opposite angles. |
| Concyclic Points | Concyclic test for four points: if one pair of opposite angles in the quadrilateral is supplementary, the four points lie on one circle. | Students may say any four points are concyclic. |
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
Try a 15-question quiz from this chapter. Get instant score and unlock concept-wise analytics.
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