Circles Class 9 Maths Formulas – All Theorems, Properties & Quick Revision Guide
Table of Contents
Home > Class 9 Maths > Circles Class 9 Maths Formulas
What Is a Circle? Key Definitions
A circle is the set of all points in a plane that are at a fixed distance from a fixed point. The fixed point is the centre and the fixed distance is the radius.
| Term | Definition |
|---|---|
| Centre | Fixed point equidistant from every point on the circle. |
| Radius (r) | Distance from centre to any point on the circle. |
| Diameter (d) | Longest chord; passes through the centre; d = 2r. |
| Circumference | Total length of the circle = 2πr. |
Understanding these basic definitions is the foundation for every theorem that follows. Students who sit JEE Main later find that coordinate geometry questions on circles trace directly back to these Class 9 ideas — so building clarity here pays off for years.
Parts of a Circle – Elements You Must Know
Chord, Diameter, and Secant
- Chord — A line segment joining any two points P and Q on the circumference. Written as PQ.
- Diameter — A chord that passes through the centre. It is the longest possible chord.
- All diameters of a circle are equal in length.
- Diameter = 2 × Radius
- Secant — A line that intersects the circle at two distinct points.
Arc, Segment, and Sector
- Arc — The portion of the circumference between two points P and Q.
- The shorter portion is the minor arc; the longer portion is the major arc.
- When PQ is a diameter, both arcs are equal, and each is called a semicircle.
- Segment — The region between a chord and an arc.
- Minor segment — region between the chord and the minor arc.
- Major segment — region between the chord and the major arc.
- Sector — The region bounded by an arc and the two radii joining the centre to the endpoints of the arc.
- Minor sector → smaller region.
- Major sector → larger region.
Tangent
A tangent is a line that touches the circle at exactly one point (the point of tangency).
- At any point on a circle, there is one and only one tangent.
- From an external point, exactly two tangents can be drawn to a circle.
All Formulas of Circles Class 9 at a Glance
This is the table students most often search for — a single-screen reference covering every key relationship in the chapter.
| Concept | Formula / Rule |
|---|---|
| Diameter | d = 2r |
| Circumference | C = 2πr |
| Area | A = πr² |
| Angle at centre vs angle at arc | ∠AOB = 2 × ∠APB (P on remaining arc) |
| Angle in a semicircle | ∠APB = 90° (when AB is diameter) |
| Angles in the same segment | ∠APB = ∠AQB (P, Q on same arc) |
| Cyclic quadrilateral (opposite angles) | ∠A + ∠C = 180°; ∠B + ∠D = 180° |
| Equal chords → equal angles at centre | If PQ = RS, then ∠POQ = ∠ROS |
| Perpendicular from centre to chord | Bisects the chord: AM = MB |
| Equal chords → equidistant from centre | If PQ = RS, then OM = ON |
India's Best Exam Preparation for Class 9th - Download Now
Elements of Circle
Let $P$ and $Q$ are two points on the circumference of the circle.
- Chord - Line Segment (PQ) joining any two points on a circle is called chord. Chord that passes through the center of the circle is called the diameter.1.Diameter is the longest chord. 2.All diameters have the same length, which is equal to two times the radius.
- Arc - A piece of a circle between two points is called Arc. The longer one is called the major arc and the shorter one is called the minorarc.1.When $P$ and $Q$ are ends of a diameter then both arcs are equal and each is called a semicircle.
- Segment - The reginn between a chord and either of its arcs is called a segment of the circle. The larger region is called as major segment and the smaller region is called as minor segment.
- Sector - The region between an arc and the two radii, joining the center to the end points of the arc is called a sector. Larger one is called as major sector and the smaller one as minor sector. 1.When two arcs are equal, that is, each is a semicircle, then both segments and both sectors become the same and each is known as a semicircular region.
- Tangent - A line which intersects the circle at exactly one point is called a tangent. 1.At a point of a circle there is one and only one tangent. 2.From an external point of a circle two tangents can be drawn to the circle.
- Secant - A line which intersects the circle at two distinct points is called a Secant.
India's Best Exam Preparation for Class 9th - Download Now
Properties of a Circle
- Two circles are congruent if and only it they have equal radii.
- Equal chords of a circle subtend equal angles at the center. Conversely if the angles subtended by the chords at the center are equal, then chords are equal.
- The perpendicular drawn from the center of a circle to a chord bisects the chord. Conversely the line drawn from the center of a circle to bisect a chord is perpendicular to the chord.
- There is one and only one circle passing through three given non - collinear points.
- Equal chords of a circle are equidistant from the center. Conversely Chords equidistant from the center of a circle are equal in length.
- If two arcs of a circle are congruent, then their corresponding chords are equal. Conversely, if two chords of a circle are equal, then their corresponding arcs are congruent.
- Congruent arcs of a circle subtends equal angle at the center.
- The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
- Angles in the same segment of a circle are equal.
- Angle in a semicircle is a right angle.
- If a line-segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line-segment, the four points lie on a circle. i.e. concyclic.
- The sum of the either pair of the opposite angles of a cyclic quadrilateral ${ }^{*}$ is $180^{\circ}$. $\angle A+\angle C=180^{\circ}$ Conversely if the sum of a pair of opposite angles of a quadrilateral is $180^{\circ}$ (i.e. supplementary), the quadrilateral is cyclic.
India's Best Exam Preparation for Class 9th - Download Now
Major vs Minor: Arc, Segment, and Sector Compared
A very common source of confusion is mixing up arc, segment, and sector. The table below fixes that.
| Element | Minor Version | Major Version | Special Case |
|---|---|---|---|
| Arc | Shorter arc PQ | Longer arc PQ | Equal arcs when PQ is the diameter (semicircle) |
| Segment | Region between the chord and the minor arc | Region between the chord and the major arc | Semicircular region when PQ is the diameter |
| Sector | Smaller pie-slice region (∠ < 180°) | Larger pie-slice region (∠ > 180°) | Each sector = semicircle when arcs are equal |
For Class 9 NCERT exercises, you will mostly work with minor arc and minor segment unless a problem specifies otherwise.
For step-by-step worked solutions to every NCERT exercise in this chapter, check out the NCERT Solutions on eSaral.
Also Read,
Download NCERT Class 9 Maths Book in PDF
Download NCERT Class 9 Science Book in PDF
Download NCERT Class 9 Science Exemplar in PDF
Download NCERT Class 9 Maths Exemplar in PDF
If you have any Confusion related to the Introduction to Circles Class 9 Maths All Formulas, then feel free to ask in the comments section down below.
To watch Free Learning Videos on Class 9 Maths by Kota’s top Faculties, install the eSaral App
Frequently Asked Questions
Find answers to common questions.
How many theorems are there in Circles Class 9 NCERT?
The NCERT Class 9 Maths Chapter on Circles covers 11 key theorems (including converses). These range from chord bisection and equidistant chords to the central angle theorem, angles in the same segment, angle in a semicircle, and the cyclic quadrilateral property.
What is the angle in a semicircle?
The angle inscribed in a semicircle is always 90°. This means if AB is the diameter of a circle and P is any point on the semicircle, then ∠APB = 90°. This is a direct result of the central angle theorem where the central angle for a diameter is 180°, and 180° ÷ 2 = 90°.
What is the formula for the angle subtended by an arc at the centre in Class 9?
The angle subtended by an arc at the centre is double the angle subtended by the same arc at any point on the remaining part of the circle. If ∠AOB is the central angle and ∠APB is the inscribed angle, then ∠AOB = 2 × ∠APB. This is the most tested theorem in the chapter.
Are equal chords equidistant from the centre?
Yes. Equal chords of a circle are always equidistant from the centre. Conversely, chords that are equidistant from the centre are equal in length. Distance from the centre to a chord means the perpendicular distance, i.e., the length of the perpendicular dropped from the centre to that chord.
What is the difference between a chord and a diameter?
A chord is any line segment joining two points on the circumference of a circle. A diameter is a special chord that passes through the centre. The diameter is the longest possible chord and equals twice the radius: d = 2r. Every diameter is a chord, but not every chord is a diameter.
What is the sum of opposite angles in a cyclic quadrilateral?
The sum of each pair of opposite angles in a cyclic quadrilateral is 180°. So if ABCD is a cyclic quadrilateral, then ∠A + ∠C = 180° and ∠B + ∠D = 180°. The converse is also true and is equally important for exam proofs.