Areas of Sector and Segment of a Circle: Formulas and Examples

In geometry, understanding sectors and segments of a circle is fundamental, as they help calculate specific portions of a circular region with precision. These subdivisions of a circle are essential in many real-world scenarios and applications, from the urban planning of circular parks and roundabouts to the manufacturing of wheels, gears, and decorative designs in architecture and engineering. This topic aims to provide a comprehensive understanding of how to compute these areas accurately, using clear formulas and practical applications that mirror real-life situations.

Mastering the calculation of areas of sectors and segments not only improves problem-solving skills but also sharpens spatial awareness. This knowledge plays a crucial role in various fields, such as engineering, architecture, and urban development, and enhances mathematical reasoning skills that are crucial for both academic and professional success.

Key Concepts

Sector of a Circle: A region formed by two radii and the arc between them. The size of the sector is determined by the central angle, measured in degrees. Sectors are classified into:
- Major Sector: Covers more than half of the circle.
- Minor Sector: Covers less than half of the circle.
Segment of a Circle: A region enclosed by a chord and the corresponding arc. It can be classified into:
- Major Segment: The larger area formed by a chord and an arc.
- Minor Segment: The smaller area formed between a chord and the corresponding arc.
Central Angle (θ): The angle formed at the center of the circle by two radii that define the sector.
Chord: A straight line connecting two points on the circumference of a circle.
Arc: A curved section of the circle’s circumference that lies between two points.
Radius (r): The distance from the center of the circle to any point on its circumference.
Circumference: The total distance around the circle, calculated using the formula: $\boldsymbol{C = 2\pi r}$

Formulas for Areas of Sector and Segment

Area of a Sector:
- Formula: $\displaystyle\boldsymbol{A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2}$
- This formula calculates the area of a sector by determining what fraction of the full circle (360°) the sector represents.
Area of a Segment:
- Formula: $\boldsymbol{A_{segment} = A_{sector} - A_{triangle}}$
- This calculates the segment area by subtracting the triangular area from the sector’s area.
Length of an Arc:
- Formula: $\displaystyle\boldsymbol{L = \frac{\theta}{360^\circ} \times 2\pi r}$
- This measures the length of the curved boundary of the sector.
Area of Triangle in a Segment:
- Formula: $\displaystyle\boldsymbol{A_{triangle} = \frac{1}{2} r^2 \sin \theta}$
- This calculates the area of a triangle formed by the two radii and the chord.
Perimeter of a Sector:
- Formula: $\displaystyle\boldsymbol{P = 2r + \frac{\theta}{360^\circ} \times 2\pi r}$
- Calculates the complete perimeter including two radii and the arc length.

Examples with Expanded Step-by-Step Solutions

Example 1: Determine the area of a sector with a radius of 14 cm and a central angle of $90^\circ$ .

Solution:

Using the formula: $\displaystyle\boldsymbol{A_{sector} = \frac{90}{360} \times \pi (14)^2 = \frac{1}{4} \times 3.14 \times 196 = 154 \text{ cm}^2}$
Explanation: Since the central angle is one-fourth of a full circle, the area of the sector is one-fourth of the total area of the circle.

Answer: $\boldsymbol{154 \text{ cm}^2}$

Example 2: Calculate the area of a segment with a radius of 10 cm and a central angle of $60^\circ$ .

Solution:

Area of Sector: $\displaystyle\boldsymbol{A_{sector} = \frac{60}{360} \times \pi (10)^2 = \frac{1}{6} \times 3.14 \times 100 = 52.36 \text{ cm}^2}$
Area of Triangle: $\displaystyle\boldsymbol{A_{triangle} = \frac{1}{2} r^2 \sin 60^\circ = 43.3 \text{ cm}^2}$
Area of Segment: $\boldsymbol{A_{segment} = 52.36 - 43.3 = 9.06 \text{ cm}^2}$

Answer: $\boldsymbol{9.06 \text{ cm}^2}$

Example 3: Calculate the length of an arc with a radius of 7 cm and a central angle of $120^\circ$ .

Solution:

Using the formula: $\displaystyle\boldsymbol{L = \frac{120}{360} \times 2\pi \times 7 = \frac{1}{3} \times 2 \times 3.14 \times 7 = 14.66 \text{ cm}}$

Answer: $\boldsymbol{14.66 \text{ cm}}$

Example 4: Find the perimeter of a sector with a radius of 8 cm and a central angle of $45^\circ$ .

Solution:

Using the perimeter formula: $\displaystyle\boldsymbol{P = 2r + \frac{\theta}{360^\circ} \times 2\pi r = 2(8) + \frac{45}{360} \times 2\pi \times 8 = 16 + 6.28 = 22.28 \text{ cm}}$

Answer: $\boldsymbol{22.28 \text{ cm}}$

Applications of Areas of Sector and Segment

Engineering: Design of machine components like gears, rotors, and mechanical discs that utilize circular segments for efficient function.
Architecture: Calculation of portions of circular structures, such as arches, domes, amphitheaters, and decorative pathways.
Agriculture: Planning circular irrigation systems or determining land areas of circular plots for better resource allocation.
Art and Design: Creating intricate patterns in visual arts, including mosaic designs and circular motifs used in graphic design.
Urban Planning: Designing roundabouts, fountains, and circular parks in city layouts for aesthetic and practical purposes.