Introduction to Circles
Introduction to circle and its parts.
What is a Circle?
A circle is one of the most important geometric shapes you will study in Class 9. Rather than thinking of it simply as a "round shape," mathematics defines it precisely: a circle is the set of all points in a plane that are at a fixed distance from a fixed point. That fixed point is called the centre, and the fixed distance is called the radius.
Circle = { P : distance(P, O) = r }In other words, every single point that lies on the boundary of a circle is exactly the same distance (the radius) from the centre. This is what makes a circle perfectly symmetric in all directions.
Three Regions Created by a Circle
When a circle is drawn on a plane, it divides the entire plane into three distinct regions. Understanding which region a point belongs to is a foundational idea tested in both Telangana and Andhra Pradesh board exams.
| Region | Description | Example Points | Condition |
|---|---|---|---|
| 🟢 Interior | Inside the circle | O, A, B, M | distance from O < r |
| 🔵 Circumference | On the circle boundary | P, G, N | distance from O = r |
| 🔴 Exterior | Outside the circle | M', Q | distance from O > r |
Chord and Diameter
A chord is any line segment whose two endpoints both lie on the circle. You can draw many different chords in a circle — they can be short or long, and they all connect two points on the circumference.
The diameter is the longest chord of any circle. It is special because it passes through the centre of the circle. Every diameter divides the circle into two equal halves.
Diameter (d) = 2 × Radius (r)If the radius of a circle is 7 cm, then its diameter = 2 × 7 = 14 cm. Conversely, if the diameter is 10 cm, the radius = 10 ÷ 2 = 5 cm.
| Term | Definition | Passes Through Centre? | Length |
|---|---|---|---|
| Chord | Line segment joining any two points on the circle | Not necessarily | Varies (less than or equal to diameter) |
| Diameter | Chord that passes through the centre | Yes — always | = 2r (longest possible chord) |
| Radius | Line segment from centre to any point on circle | Starts from centre | = d / 2 |
Arcs — Minor, Major and Semicircle
When two points lie on a circle, they divide the circle's boundary into two curved pieces. Each piece is called an arc. The arc between points A and B is written as arc AB (with a small curve symbol above AB).
Segments of a Circle
When a chord is drawn inside a circle, it divides the circular region into two parts called segments. Think of a chord cutting a pizza — you get two unequal pieces (unless the chord is the diameter).
| Type of Segment | How It's Formed | Size |
|---|---|---|
| Major Segment | The larger region between a chord and the major arc | Bigger than half the circle |
| Minor Segment | The smaller region between a chord and the minor arc | Smaller than half the circle |
| Semicircle | When the chord is a diameter | Exactly half the circle |
Sectors of a Circle — Minor and Major
A sector is a "pie-slice" shaped region formed by two radii and the arc between them. Imagine cutting a circular pizza — each slice is a sector. The region is enclosed by two straight edges (radii) and one curved edge (the arc).
A sector is bounded by two radii + one arc (like a pie slice with the pointed centre included).
A segment is bounded by one chord + one arc (like a pie slice with the pointed tip cut off flat).
Complete Vocabulary at a Glance
Here is every key term from the Chapter 12 introduction, clearly defined and ready for quick revision before your CBSE, Telangana, or AP board exam:
Common Mistakes to Avoid
- Confusing sector and segment: A sector includes the two radii (pointed centre-piece of a pie). A segment does NOT include the centre — it is cut off by a chord. Draw both and compare.
- Calling all chords "diameters": Only the chord that passes through the centre is a diameter. Other chords are shorter and do not have this special property.
- Mixing up minor and major: "Minor" always refers to the smaller part (arc or segment or sector). "Major" always refers to the larger part.
- Forgetting that diameter = 2 × radius: If a question gives you the radius, double it for the diameter. If it gives the diameter, halve it for the radius.
- Confusing "circumference" with "area": Circumference is the length of the boundary (the curved line). It is not the area of the circle's interior.
What This Lesson Prepares You For
This introduction to circles builds the vocabulary foundation for the rest of Chapter 12, where you will study important theorems — such as the relationship between a chord's length and its distance from the centre, the angle subtended by a chord, and cyclic quadrilaterals. Every theorem uses the terms introduced here, so mastering these definitions now saves significant effort later.
The concept of arcs and segments connects directly to mensuration topics (areas of sectors and segments) that appear in higher classes. The chord-diameter relationship also reappears when you study the Triangles chapter (especially the angle in a semicircle theorem). For Class 8 revision, your earlier work on Construction of Quadrilaterals used compass constructions that are now formalised through circle geometry.