Arc and Sector of a Circle

A circle is a closed curve where every point is at the same distance from the centre. An arc is a part of the circumference of a circle, while a sector is the region enclosed between two radii and an arc. An arc and a sector of a circle are fundamental concepts in geometry that help in measuring the parts of a circle. Understanding their formulas makes solving problems involving angles and lengths much easier. In this guide, you will learn the definition of an arc and a segment of a circle, the relation between them and solved problems for easy understanding.

Table of Contents

• What is an Arc and a Sector of a Circle
• Types and Properties of an Arc and a Sector of a Circle
• Relationship between Arc and Sector of a Circle
• Real-World Applications of an Arc and a Sector of a Circle
• Solved Examples on an Arc and a Sector of a Circle

What is an Arc and a 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 let us learn about two important parts of a circle: the arc and the sector.

• Arc of a Circle: An arc is any part of the circumference (boundary) of a circle.

• Sector of a Circle: A sector is the region enclosed between two radii and the arc connecting their endpoints.

Types and Properties of an Arc and a Sector of a Circle 

Major and minor arcs:

Let A and B be two points on a circle; they divide the circumference (boundary) into two arcs:

• Minor arc (AB): Minor arc AB is the shorter part of the circumference.

• Major arc (AB): Major arc AB is the longer part of the circumference.

• A minor arc + a major arc = the full circumference of the circle.

Major and minor segments:

Let A and B be two points on a circle; they divide the circumference (boundary) into two arcs:

• Minor sector: A minor sector is the region enclosed between two radii and the minor arc connecting their endpoints. The smaller slice of the circle. Its central angle is less than 180°.

• Major sector: A major sector is the region enclosed between two radii and the major arc connecting their endpoints. The larger slice of the circle. Its central angle is more than 180°.

• Minor sector + major sector = full circle.

• Quadrant: The sector with a central angle of 90° is called a quadrant (one-quarter of the circle).

• Semicircle: The sector with central angle = 180° is called a semicircle (one-half of the circle).

Relationship between Arc and Sector of a Circle

An arc and its corresponding sector are both determined by the same central angle (θ) and radius (r) of a circle.
The relationship between an arc and a sector is the following:

Area of a sector =   1/2× r × l

Where r = radius and l = arc length.

This formula summarises that:

• The area of a sector is directly proportional to its arc length.

• If the arc length increases, the sector area also increases (for the same radius). 

Real-World Applications of an Arc and a Sector of a Circle

• Pie Charts: Each slice of a pie chart is a sector, where the central angle represents what fraction of the whole a category makes up. 

• Clock Faces: The hands of a clock sweep out sectors as time passes. The minute hand travels the full 360° every hour.

• Pizza and Cake Slices: Cutting a circular pizza into equal slices produces equal sectors, resulting in the same area in every piece.

• Windscreen Wipers: A car's windscreen wiper sweeps back and forth across a sector-shaped area, clearing only that portion of the glass.

Solved Examples on an Arc and a Sector of a Circle

Example 1: Two points, P and Q, are on a circle. The minor arc PQ is 7 cm, and the total circumference is 23 cm. Find the major arc PQ.
Solution: Circumference of the circle = 23 cm
Length of the minor arc = 7 cm.
Length of major arc = Circumference - length of minor arc = 23 - 7 = 16 cm.

Example 2: A pizza is cut into 8 equal slices. What is the central angle of each slice?
Solution: A pizza is a full circle. ∴ Central angle of each = 360°. It is cut into 8 equal slices.
∴ The centre of each slice = 360° ÷ 8 = 45°.

Example 3: A sector has a central angle of 120°. What fraction of the circle does it cover?
Solution: The central angle of the sector is 120°.
360° ÷ 120° = 3. ∴ the sector is ⅓ of the entire circle.