Area of Circular Sectors and Segments

Introduction

Key Concepts

Understanding Circular Sectors

A circular sector is a portion of a circle enclosed by two radii and an arc. It resembles a 'slice of pie' and is characterized by its central angle, which is the angle formed by the two radii. The area of a circular sector is directly proportional to its central angle.

Formula for the Area of a Circular Sector:

$$ \text{Area} = \frac{\theta}{360} \times \pi r^2 $$

Where:

• θ = Central angle in degrees
• r = Radius of the circle

Example:

Find the area of a sector with a central angle of 60° in a circle with a radius of 10 cm.

Solution:

$$ \text{Area} = \frac{60}{360} \times \pi \times 10^2 = \frac{1}{6} \times \pi \times 100 \approx 52.36 \text{ cm}^2 $$

Exploring Circular Segments

A circular segment is the region of a circle between a chord and the corresponding arc. Unlike sectors, segments do not include the center of the circle. Calculating the area of a segment involves subtracting the area of the triangular portion from the area of the sector.

Formula for the Area of a Circular Segment:

$$ \text{Area} = \frac{\theta}{360} \times \pi r^2 - \frac{1}{2} r^2 \sin(\theta) $$

Where:

• θ = Central angle in degrees
• r = Radius of the circle

Example:

Calculate the area of a segment with a central angle of 90° in a circle with a radius of 8 cm.

Solution:

$$ \text{Area} = \frac{90}{360} \times \pi \times 8^2 - \frac{1}{2} \times 8^2 \times \sin(90°) = \frac{1}{4} \times \pi \times 64 - \frac{1}{2} \times 64 \times 1 $$ $$ \text{Area} \approx 50.27 \text{ cm}^2 - 32 \text{ cm}^2 = 18.27 \text{ cm}^2 $$

Derivation of Formulas

The area formulas for sectors and segments are derived from the basic properties of circles and trigonometry. For a sector, the area is a fraction of the total area of the circle, with the fraction determined by the central angle. For segments, the area is the difference between the sector and the triangular area formed by the radii and the chord.

Sector Area Derivation:

The total area of a circle is $\pi r^2$. A sector with a central angle θ degrees occupies $\frac{\theta}{360}$ of the circle.

$$ \text{Area} = \frac{\theta}{360} \times \pi r^2 $$

Segment Area Derivation:

The segment area is obtained by subtracting the area of the triangle formed by the two radii and the chord from the area of the sector.

$$ \text{Area of Triangle} = \frac{1}{2} r^2 \sin(\theta) $$ $$ \text{Area of Segment} = \frac{\theta}{360} \times \pi r^2 - \frac{1}{2} r^2 \sin(\theta) $$

Applications of Sectors and Segments

Circular sectors and segments have widespread applications in various fields:

• Engineering: Designing gears and rotational components.
• Architecture: Creating arch structures and domes.
• Art and Design: Crafting circular motifs and patterns.
• Astronomy: Measuring angles and areas on celestial bodies.

Exercises and Practice Problems

To reinforce the understanding of circular sectors and segments, consider solving the following problems:

1. Find the area of a sector with a central angle of 120° in a circle with a radius of 15 cm.
2. Calculate the area of a segment with a central angle of 45° in a circle of radius 12 cm.
3. Derive the formula for the area of a circular segment when the central angle is given in radians.
4. A circular garden has a central angle of 80°. If the radius is 10 meters, determine the length of the arc.

Solving Practice Problems

Problem 1: Find the area of a sector with a central angle of 120° in a circle with a radius of 15 cm.

Solution:

$$ \text{Area} = \frac{120}{360} \times \pi \times 15^2 = \frac{1}{3} \times \pi \times 225 = 75\pi \approx 235.62 \text{ cm}^2 $$

Problem 2: Calculate the area of a segment with a central angle of 45° in a circle of radius 12 cm.

Solution:

$$ \text{Area of Sector} = \frac{45}{360} \times \pi \times 12^2 = \frac{1}{8} \times \pi \times 144 = 18\pi \approx 56.55 \text{ cm}^2 $$ $$ \text{Area of Triangle} = \frac{1}{2} \times 12^2 \times \sin(45°) = \frac{1}{2} \times 144 \times \frac{\sqrt{2}}{2} = 36\sqrt{2} \approx 50.91 \text{ cm}^2 $$ $$ \text{Area of Segment} = 18\pi - 36\sqrt{2} \approx 56.55 - 50.91 = 5.64 \text{ cm}^2 $$

Advanced Concepts

For students aiming to delve deeper, understanding segments and sectors in polar coordinates and integrating to find areas can provide a more comprehensive grasp of the topic. Additionally, exploring the relationships between radians and degrees in central angles enhances problem-solving flexibility.

Comparison Table

Aspect	Circular Sector	Circular Segment
Definition	A region bounded by two radii and an arc.	A region bounded by a chord and an arc.
Area Formula	$\frac{\theta}{360} \times \pi r^2$	$\frac{\theta}{360} \times \pi r^2 - \frac{1}{2} r^2 \sin(\theta)$
Includes Center	Yes	No
Common Applications	Designing pie charts, sectors in engineering.	Architectural arches, lens shapes in optics.
Pros	Simpler to calculate, widely applicable.	Useful in areas requiring precise segmentation.
Cons	Less useful when the center is not involved.	Requires additional calculations for the triangular area.

Summary and Key Takeaways