Arc and Sector of a Circle
You already know that a circle is a closed curve where every point is at the same distance from the centre. Now it is time to learn about two important parts of a circle — the arc and the sector.
An arc is a part of the boundary (circumference) of a circle. A sector is the region (area) enclosed between two radii and an arc — like a slice of pizza or a piece of pie.
Understanding arcs and sectors is useful in everyday life — clock faces, pie charts, pizza slices, and fan blades are all examples of sectors.
What is Arc and Sector of a Circle?
Definition: An arc is any part of the circumference (boundary) of a circle.
Definition: A sector is the region enclosed between two radii and the arc connecting their endpoints. It looks like a slice of a circular cake.
Key terms:
- Minor arc: The shorter arc between two points on a circle.
- Major arc: The longer arc between the same two points.
- Minor sector: The smaller region (slice) enclosed by two radii and the minor arc.
- Major sector: The larger region enclosed by two radii and the major arc.
- Central angle: The angle formed at the centre of the circle by the two radii. It decides how big the arc and sector are.
- Semicircle: When the two radii form a straight line (central angle = 180°), each sector is a semicircle.
Types and Properties
1. Minor Arc and Major Arc
When two points A and B are on a circle, they divide the circumference into two arcs:
- Minor arc (AB): The shorter part of the circumference.
- Major arc (AB): The longer part of the circumference.
- Minor arc + Major arc = Full circumference.
- If both arcs are equal, each is a semicircular arc.
2. Minor Sector and Major Sector
- Minor sector: The smaller slice. Its central angle is less than 180°.
- Major sector: The larger slice. Its central angle is more than 180°.
- Minor sector + Major sector = Full circle.
3. Special Sectors
- Quadrant: Central angle = 90° (one-quarter of the circle).
- Semicircle: Central angle = 180° (one-half of the circle).
- Full circle: Central angle = 360°.
4. Relationship between Arc and Sector
- Every sector has an arc as its curved boundary.
- A bigger central angle means a longer arc and a larger sector.
- The arc is part of the circumference; the sector is part of the area.
Solved Examples
Example 1: Example 1: Identifying arcs
Problem: Two points P and Q are on a circle. The minor arc PQ is 5 cm and the total circumference is 20 cm. Find the major arc PQ.
Solution:
- Major arc = Circumference − Minor arc
- Major arc = 20 − 5 = 15 cm
Answer: The major arc PQ is 15 cm.
Example 2: Example 2: Pizza slices as sectors
Problem: A pizza is cut into 8 equal slices. What is the central angle of each slice?
Solution:
- Full circle = 360°.
- Each slice = 360° ÷ 8 = 45°.
- Since 45° < 180°, each slice is a minor sector.
Answer: Each slice has a central angle of 45° and is a minor sector.
Example 3: Example 3: Quadrant as a sector
Problem: What fraction of a circle is a quadrant? What is its central angle?
Solution:
- A quadrant is one-fourth of a circle.
- Central angle = 360° ÷ 4 = 90°.
Answer: A quadrant is 1/4 of a circle with a central angle of 90°.
Example 4: Example 4: Semicircle
Problem: A circle has a diameter drawn through it. What sectors are formed? What is the arc length of each if the circumference is 44 cm?
Solution:
- A diameter divides a circle into two equal semicircles.
- Central angle of each = 180°.
- Each semicircular arc = 44 ÷ 2 = 22 cm.
Answer: Two semicircles are formed. Each has an arc length of 22 cm.
Example 5: Example 5: Clock face as sectors
Problem: On a clock, the minute hand moves from 12 to 3. What sector is traced? What is the central angle?
Solution:
- From 12 to 3 is one-quarter of the clock face.
- Central angle = 360° ÷ 4 = 90°.
- The region swept is a quadrant.
Answer: A quadrant with a central angle of 90°.
Example 6: Example 6: Finding the central angle
Problem: A sector is 1/6 of a circle. What is its central angle?
Solution:
- Central angle = (1/6) × 360° = 60°.
Answer: The central angle is 60°.
Example 7: Example 7: Finding the fraction from the angle
Problem: A sector has a central angle of 120°. What fraction of the circle does it cover?
Solution:
- Fraction = 120° ÷ 360° = 1/3.
Answer: The sector covers 1/3 of the circle.
Example 8: Example 8: Major sector angle
Problem: A minor sector has a central angle of 80°. What is the central angle of the major sector?
Solution:
- Major sector angle = 360° − 80° = 280°
Answer: The major sector has a central angle of 280°.
Example 9: Example 9: Pie chart as sectors
Problem: In a pie chart, the sector for "Cricket" has a central angle of 90° and "Football" has 60°. Which sport is more popular?
Solution:
- Cricket: 90° ÷ 360° = 1/4 (25%).
- Football: 60° ÷ 360° = 1/6 (about 16.7%).
- Cricket has a larger angle, so more students prefer cricket.
Answer: Cricket is more popular.
Example 10: Example 10: Number of sectors
Problem: A circle is divided into sectors with central angles of 60° each. How many sectors are formed?
Solution:
- Number of sectors = 360° ÷ 60° = 6.
Answer: 6 equal sectors are formed.
Real-World Applications
Pie Charts: Each slice of a pie chart is a sector. The central angle shows what fraction of the total each category represents.
Clock Faces: The hands of a clock sweep out sectors as they move. The minute hand completes 360° every hour.
Pizza and Cake Slices: Cutting a circular pizza into equal slices creates equal sectors.
Fans and Windshield Wipers: A ceiling fan blade traces a sector. A car windshield wiper sweeps a sector-shaped area.
Protractors: A protractor is a semicircular tool (180° sector) used to measure angles.
Key Points to Remember
- An arc is a part of the circumference of a circle.
- A sector is the region between two radii and an arc — like a pizza slice.
- Minor arc is the shorter arc; major arc is the longer arc.
- Minor sector has a central angle less than 180°.
- Major sector has a central angle more than 180°.
- Minor arc + Major arc = Full circumference.
- Minor sector + Major sector = Full circle.
- A quadrant is a sector with central angle 90°.
- A semicircle is a sector with central angle 180°.
- Fraction of circle = Central angle ÷ 360°.
Practice Problems
- A circle has minor arc AB = 8 cm and circumference = 30 cm. Find the major arc AB.
- A sector has a central angle of 72°. What fraction of the circle does it represent?
- A pie chart has 5 equal sectors. What is the central angle of each sector?
- The minor sector has a central angle of 110°. Find the central angle of the major sector.
- A clock's minute hand moves from 12 to 6. What central angle does it sweep?
- How many sectors of 40° each can fit in a full circle?
- A pizza is cut into 6 equal slices. What is the central angle of each slice?
- Draw a circle and shade a sector with central angle 120°. Is it a minor or major sector?
Frequently Asked Questions
Q1. What is the difference between an arc and a sector?
An arc is a curved line (part of the circumference). A sector is a region (area) enclosed between two radii and an arc. The arc is the crust edge of a pizza slice, and the sector is the entire slice.
Q2. What is the difference between a minor arc and a major arc?
When two points are on a circle, the shorter arc between them is the minor arc and the longer one is the major arc. Together they make the full circumference.
Q3. How is a sector different from a segment?
A sector is between two radii and an arc (like a pizza slice). A segment is between a chord and the arc it cuts off. At Class 6 level, you mainly work with sectors.
Q4. What is a quadrant?
A quadrant is a sector with a central angle of 90°. It is one-fourth of a circle.
Q5. How do you find the central angle of a sector?
If you know the fraction of the circle, multiply that fraction by 360°. For example, 1/3 of a circle = (1/3) × 360° = 120°.
Q6. Can a sector have a central angle of exactly 180°?
Yes. A sector with a 180° central angle is a semicircle — exactly half of the circle.
Q7. Where do we see arcs and sectors in real life?
Pie charts, clock faces, pizza slices, fan blades, windshield wipers, protractors, and circular gardens with pathways all involve arcs and sectors.
Q8. What is a semicircular arc?
When two points on a circle are endpoints of a diameter, the two arcs formed are equal. Each arc is a semicircular arc — exactly half the circumference.
Related Topics
- Circle - Basic Concepts
- Area of Sector of a Circle
- Arc Length of a Circle
- Circumference of 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
