Sector And Segment Of A Circle

Understanding the concepts of a sector and a segment of a circle is essential in both geometry and practical applications such as engineering, architecture, and even art. These two terms describe specific portions of a circle, but they are not the same. A sector of a circle is like a slice of pie,” while a segment of a circle is more like a “cut-off portion” defined by a chord. Both play an important role in calculating areas, angles, and arc lengths, and they appear in a wide range of real-world problems. By exploring their definitions, properties, and formulas, you can master how to distinguish and work with them effectively.

Defining the Sector of a Circle

A sector of a circle is the region enclosed by two radii and the arc between them. It resembles a wedge or slice of a pizza, where the tip is at the circle’s center. The size of a sector depends on the central angle, which is the angle between the two radii.

Key Properties of a Sector

• Bounded by two radii and an arc.
• The central angle determines its size.
• The larger the central angle, the larger the sector area.
• Measured in degrees or radians.

Formula for the Area of a Sector

The formula to find the area of a sector is

Area of sector = (θ / 360°) à πr² (when θ is in degrees)

or

Area of sector = (θ / 2) à r² (when θ is in radians)

Where

• θ = central angle
• r = radius of the circle

Example

If a circle has a radius of 10 cm and the central angle is 90°, the area of the sector is

(90 / 360) à π à 10² = 1/4 à π à 100 = 25π cm²

Defining the Segment of a Circle

A segment of a circle is the region bounded by a chord and the arc lying between the chord’s endpoints. Unlike a sector, a segment does not include the center of the circle unless it is a semicircle.

Key Properties of a Segment

• Formed by a chord and its corresponding arc.
• Can be a minor segment (smaller area) or a major segment (larger area).
• The size of the segment depends on the chord length and arc length.

Formula for the Area of a Segment

The area of a segment is found by subtracting the area of the triangle formed by the chord and the radii from the area of the sector that contains the segment.

Area of segment = Area of sector − Area of triangle

Example

If a circle has a radius of 10 cm and a central angle of 60°, the area of the segment can be found by

• Area of sector = (60 / 360) à π à 10² = (1/6) à π à 100 ≈ 52.36 cm²
• Area of triangle = (1/2) à r² à sin(θ) = 0.5 à 100 à sin(60°) ≈ 43.30 cm²
• Area of segment ≈ 52.36 − 43.30 ≈ 9.06 cm²

Difference Between Sector and Segment

While both are parts of a circle, their boundaries and definitions differ significantly.

Comparison Table

• SectorFormed by two radii and an arc.
• SegmentFormed by a chord and an arc.
• SectorAlways includes the center of the circle.
• SegmentDoes not necessarily include the center.
• SectorMeasured using central angle directly.
• SegmentRequires subtraction of triangle area from sector area.

Applications in Real Life

Both sectors and segments are widely used in real-world contexts

• EngineeringGear and wheel designs often involve circular sectors for calculating rotation and torque.
• ArchitectureDome and arch designs require segment calculations for material usage.
• DesignCreating fan blades, pie charts, or curved furniture pieces involves sector geometry.
• Land MeasurementCircular plots and curved boundaries often require sector and segment analysis.

Arc Length and Its Connection

Both sector and segment calculations often involve finding the arc length, which is the distance along the curved part of the circle. The formula is

Arc length = (θ / 360°) à 2πr (degrees)

or

Arc length = θ à r (radians)

Example of Arc Length

If a circle has a radius of 8 m and the central angle is 45°, the arc length is

(45 / 360) à 2π à 8 = (1/8) à 16π = 2π m

Minor and Major Sectors and Segments

In any circle, there can be a smaller portion (minor) and a larger portion (major) for both sectors and segments.

Minor Sector

Central angle less than 180°, smaller than a semicircle.

Major Sector

Central angle greater than 180°, larger than a semicircle.

Minor Segment

Smaller portion cut off by the chord.

Major Segment

Larger portion left after the chord cuts off the minor segment.

Steps to Solve Problems

• Identify whether the problem involves a sector or a segment.
• Find the radius and central angle of the circle.
• Apply the relevant formulas for area or arc length.
• If it’s a segment, subtract the triangle’s area from the sector’s area.
• Check units and round off the answer if necessary.

Practice Problems

• Find the area of a sector with radius 12 cm and central angle 120°.
• Calculate the arc length of a circle with radius 15 cm and central angle 72°.
• Determine the area of a segment where radius is 9 cm and central angle is 60°.
• Find the difference between the areas of the major and minor sectors of a circle with radius 20 cm and central angle 100°.

Understanding the sector and segment of a circle is fundamental to mastering geometry. Sectors are defined by two radii and an arc, while segments are defined by a chord and an arc. By knowing the difference, applying the correct formulas, and practicing with examples, you can handle problems involving these shapes in both academic and practical settings. Whether in construction, design, or mathematics, this knowledge makes calculations involving circular shapes accurate and efficient.