Circle - Basic Concepts
Look around you — the wheel of a bicycle, a coin, a bangle, a clock face, a pizza. All these objects have a circular shape. A circle is one of the most common shapes in nature and daily life.
A circle is a closed curve where every point on the curve is at the same distance from a fixed point called the centre. Unlike polygons (triangles, squares), a circle has no straight sides, no corners, and no angles.
In NCERT Class 6 Mathematics, circles are introduced in the chapter Basic Geometrical Ideas. You will learn the names and meanings of the parts of a circle — centre, radius, diameter, chord, arc, sector, and segment.
Understanding these basic terms is essential because they appear again in higher classes when you learn to calculate the circumference and area of a circle.
What is Circle - Basic Concepts?
Definition: A circle is a closed curve in a plane such that every point on the curve is at a fixed distance from a fixed point.
Key terms:
- Centre (O): The fixed point inside the circle from which all points on the circle are equidistant. The centre is NOT on the circle — it is inside it.
- Radius (r): The distance from the centre to any point on the circle. All radii (plural of radius) of a circle are equal.
- Diameter (d): A line segment that passes through the centre and has both endpoints on the circle. Diameter = 2 × Radius.
- Chord: A line segment whose both endpoints lie on the circle. A chord does NOT necessarily pass through the centre.
- Arc: A part (piece) of the circle. A minor arc is the shorter arc; a major arc is the longer arc.
- Semicircle: Half of the circle. A diameter divides a circle into two semicircles.
- Sector: The region enclosed between two radii and the arc between them (like a pizza slice).
- Segment: The region between a chord and the arc it cuts off.
Important:
- The diameter is the longest chord of a circle.
- All radii of the same circle are equal.
- Infinite radii and infinite chords can be drawn in a circle.
- There is only one centre in a circle.
Circle - Basic Concepts Formula
Key relationships:
Diameter = 2 × Radius
Radius = Diameter ÷ 2
Where:
- d = diameter (the full distance across the circle through the centre)
- r = radius (the distance from the centre to the edge)
Related formulas (studied in Class 7):
- Circumference (perimeter of circle) = 2πr = πd
- Area of circle = πr²
Units:
- Radius and diameter are measured in cm, m, mm, etc.
- If radius = 5 cm, then diameter = 10 cm.
Types and Properties
Parts of a circle and their types:
1. Radius:
- Line segment from centre to any point on the circle.
- A circle has infinitely many radii.
- All radii of the same circle are equal in length.
2. Diameter:
- A chord that passes through the centre.
- The longest chord of the circle.
- A circle has infinitely many diameters.
- All diameters of the same circle are equal.
3. Chord:
- A line segment with both endpoints on the circle.
- A chord may or may not pass through the centre.
- A diameter is a special chord (the longest one).
4. Arc:
- Minor arc: The shorter arc between two points on the circle.
- Major arc: The longer arc between the same two points.
- When the arc is exactly half the circle, it is a semicircular arc.
5. Sector:
- Minor sector: The smaller "pizza slice" region.
- Major sector: The larger region.
6. Segment:
- Minor segment: The smaller region between a chord and the minor arc.
- Major segment: The larger region between the chord and the major arc.
Solved Examples
Example 1: Example 1: Finding the diameter
Problem: The radius of a circle is 7 cm. Find the diameter.
Solution:
Given:
- Radius (r) = 7 cm
Using the formula:
- Diameter = 2 × Radius
- Diameter = 2 × 7
- Diameter = 14 cm
Answer: The diameter is 14 cm.
Example 2: Example 2: Finding the radius from diameter
Problem: The diameter of a circular coin is 2.4 cm. Find the radius.
Solution:
Given:
- Diameter (d) = 2.4 cm
Using the formula:
- Radius = Diameter ÷ 2
- Radius = 2.4 ÷ 2
- Radius = 1.2 cm
Answer: The radius is 1.2 cm.
Example 3: Example 3: Identifying parts of a circle
Problem: In a circle with centre O, a line segment AB passes through O, and a line segment CD does not pass through O. Both AB and CD have their endpoints on the circle. Identify AB and CD.
Solution:
- AB has both endpoints on the circle and passes through the centre O. Therefore, AB is a diameter.
- CD has both endpoints on the circle but does not pass through the centre. Therefore, CD is a chord.
Note: A diameter is a special type of chord that passes through the centre.
Answer: AB is a diameter; CD is a chord.
Example 4: Example 4: Radius and chord comparison
Problem: A circle has a radius of 5 cm. Can a chord of this circle be 11 cm long?
Solution:
Given:
- Radius = 5 cm
- Diameter = 2 × 5 = 10 cm
Analysis:
- The diameter (10 cm) is the longest possible chord.
- No chord can be longer than the diameter.
- 11 cm > 10 cm
Answer: No, a chord of 11 cm is not possible. The maximum chord length is the diameter = 10 cm.
Example 5: Example 5: Counting radii and diameters
Problem: How many radii can be drawn in a circle? How many diameters?
Solution:
- A radius connects the centre to any point on the circle.
- There are infinitely many points on the circle.
- Therefore, infinitely many radii can be drawn.
- A diameter connects two points on the circle through the centre.
- Since there are infinitely many points on the circle, infinitely many diameters can be drawn.
Answer: Infinitely many radii and infinitely many diameters can be drawn.
Example 6: Example 6: Identifying arcs
Problem: Two points A and B are on a circle. The shorter curve from A to B is 4 cm, and the longer curve from A to B is 12 cm. Identify the minor arc, major arc, and the total circumference.
Solution:
- Minor arc AB = shorter curve = 4 cm
- Major arc AB = longer curve = 12 cm
- Circumference = minor arc + major arc = 4 + 12 = 16 cm
Answer: Minor arc = 4 cm, Major arc = 12 cm, Circumference = 16 cm.
Example 7: Example 7: Sector identification
Problem: A pizza of radius 15 cm is cut into 6 equal slices. What shape is each slice? What is the name of this region in geometry?
Solution:
- Each slice is bounded by two radii (the straight cuts from the centre) and an arc (the curved crust).
- This region is called a sector of the circle.
- Since the pizza is divided into 6 equal parts, each sector has an angle of 360° ÷ 6 = 60° at the centre.
Answer: Each slice is a sector with a central angle of 60°.
Example 8: Example 8: Diameter vs chord
Problem: In a circle of radius 10 cm, a chord is 16 cm long. Is this chord a diameter?
Solution:
Given:
- Radius = 10 cm
- Diameter = 2 × 10 = 20 cm
- Chord = 16 cm
Analysis:
- A diameter is 20 cm long.
- The chord is 16 cm, which is less than 20 cm.
- Since the chord is shorter than the diameter, it does NOT pass through the centre.
Answer: No, the chord is not a diameter. It is 16 cm, while the diameter is 20 cm.
Example 9: Example 9: Concentric circles
Problem: Two circles have the same centre O. The inner circle has radius 3 cm and the outer circle has radius 5 cm. What are these circles called? Find the width of the ring between them.
Solution:
- Circles with the same centre are called concentric circles.
- Width of the ring = Outer radius − Inner radius
- = 5 − 3 = 2 cm
Answer: They are concentric circles. The ring width is 2 cm.
Example 10: Example 10: Segment of a circle
Problem: A chord AB divides a circle into two parts. Name these parts.
Solution:
- A chord divides the circular region into two parts.
- Each part is called a segment of the circle.
- The smaller part is the minor segment.
- The larger part is the major segment.
Note: A segment is the region between a chord and the arc. Do not confuse it with a sector (which is bounded by two radii and an arc).
Answer: The two parts are the minor segment and the major segment.
Real-World Applications
Real-world examples of circles and their parts:
- Wheels: The hub is the centre, the spoke is a radius, and the outer rim forms the circle. The axle passes through the centre.
- Clocks: The clock face is circular. The hands rotate like radii. The numbers are placed on the circle.
- Pizza: Cutting a pizza from the centre creates sectors. The crust is an arc.
- Bangles: A bangle is a circle. Its thickness is the difference between outer and inner radii (concentric circles).
- Sports: Cricket fielding circles, Olympic rings, archery targets — all use concentric circles.
- Wells and pipes: The cross-section of a pipe is a circular ring (two concentric circles). The diameter determines the pipe size.
- Ferris wheel: Passengers sit on the circle. The distance from the centre is the radius. The total span is the diameter.
Key Points to Remember
- A circle is a closed curve where every point is equidistant from a fixed point called the centre.
- The radius is the distance from the centre to any point on the circle. All radii are equal.
- The diameter passes through the centre and is the longest chord. Diameter = 2 × Radius.
- A chord is a line segment with both endpoints on the circle. It may or may not pass through the centre.
- An arc is a part of the circle. Minor arc (shorter) and major arc (longer).
- A semicircle is half the circle, formed when a diameter divides the circle.
- A sector is the region between two radii and an arc (like a pizza slice).
- A segment is the region between a chord and the arc.
- Concentric circles share the same centre but have different radii.
- A circle has infinitely many radii, diameters, and chords.
Practice Problems
- The radius of a circle is 9 cm. Find the diameter.
- The diameter of a circle is 30 cm. Find the radius.
- A circle has a radius of 6 cm. Can you draw a chord of length 13 cm in this circle? Explain.
- Name the longest chord of a circle. What is special about it?
- What is the difference between a chord and a diameter?
- A circular garden has a radius of 14 m. A straight path connects two points on the boundary, passing through the centre. How long is the path?
- How is a sector different from a segment of a circle?
- Draw a circle and mark on it: centre, one radius, one diameter, one chord (not a diameter), the minor arc, and the major arc.
Frequently Asked Questions
Q1. What is a circle?
A circle is a closed curve in a plane where every point on the curve is at the same distance from a fixed point called the centre. The fixed distance is called the radius.
Q2. What is the difference between radius and diameter?
The radius is the distance from the centre to any point on the circle. The diameter is the distance across the circle through the centre — it connects two points on the circle passing through the centre. Diameter = 2 × Radius.
Q3. Is a diameter a chord?
Yes. A diameter is a special chord — it is a chord that passes through the centre of the circle. It is the longest possible chord.
Q4. How many radii can a circle have?
Infinitely many. A radius can be drawn from the centre to any point on the circle, and there are infinitely many points on the circle.
Q5. What is the difference between an arc and a chord?
An arc is a curved part of the circle (a piece of the boundary). A chord is a straight line segment connecting two points on the circle. An arc and a chord connect the same two points, but the arc is curved and the chord is straight.
Q6. What is the difference between a sector and a segment?
A sector is the region bounded by two radii and an arc (like a pizza slice). A segment is the region between a chord and the arc. A sector always includes the centre; a segment may not.
Q7. What are concentric circles?
Concentric circles are two or more circles that have the same centre but different radii. An archery target is a real-life example of concentric circles.
Q8. Is the centre of a circle on the circle?
No. The centre is a point inside the circle, not on the circle. It is equidistant from all points on the circle.
Related Topics
- Circumference of Circle
- Area of Circle
- Arc and Sector of a Circle
- Tangent to a Circle
- Point, Line Segment, Line and Ray
- Collinear and Non-Collinear Points
- Intersecting and Parallel Lines
- Curves - Open and Closed
- Introduction to Polygons
- Introduction to Triangles
- Quadrilateral Basics
- Planes in Geometry
- Basic Geometrical Ideas
- Types of Lines
