How to Find Area and Perimeter of a Sector with Formula & Examples
The concept of sector of a circle plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding how to calculate the area and perimeter of a sector helps students solve geometry problems confidently and is highly relevant for board exams, Olympiads, and competitive tests. Let's explore the definition, types, formulas, solved examples, and quick tricks for mastering the sector of a circle.
What Is Sector of a Circle?
A sector of a circle is a portion of a circle enclosed by two radii and their intercepted arc. Imagine slicing a pizza—each slice is a sector! You’ll find this concept applied in areas such as pie charts, speedometer dials, and circular fields. In diagrams, a sector is marked by a central angle (θ), two straight radii, and the curved edge (arc).
Key Formula for Sector of a Circle
Here’s the standard formula for the area and perimeter of a sector where the central angle is θ (in degrees) and the radius is r:
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Area of a sector: \( \frac{\theta}{360^{\circ}} \times \pi r^2 \)
(If θ is in radians: \( \frac{1}{2} r^2 \theta \))
- Arc length: \( \frac{\theta}{360^{\circ}} \times 2\pi r \)
- Perimeter of sector: \( 2r + \text{arc length} \)
Types of Sector: Major and Minor
| Type | Central Angle | Portion of Circle |
|---|---|---|
| Minor Sector | Less than 180° | Smaller part |
| Major Sector | More than 180° | Larger part |
Step-by-Step Illustration: How to Find the Area of a Sector
- Start with the given: Suppose the radius (r) = 7 units and angle θ = 40°.
- Apply the sector formula: Area = \( \frac{40}{360} \times \pi \times 7^2 \)
- Calculate: \( = \frac{1}{9} \times \frac{22}{7} \times 49 \) (using π ≈ 22/7)
- Solve: \( = \frac{1}{9} \times 154 = 17.11 \) square units
Speed Trick or Vedic Shortcut
Here's a quick shortcut to calculate arc length in circle sector problems for exams. Always remember: Arc length is simply the fractional part of the circumference matching the angle.
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Example: If θ = 60°, r = 10 cm:
Fraction of circle: 60°/360° = 1/6
Arc length: (1/6) × 2πr = (1/6) × 2 × 3.14 × 10 = 10.47 cm
Fast memorization: "Take angle, divide by 360, multiply by 2πr.” Vedantu sessions teach many such smart approaches for exam speed.
Relation to Other Concepts
The idea of sector of a circle connects closely with topics such as area of a circle, segment of a circle, and circumference. Mastering sectors helps you work with fractions of geometric areas and prepares you for advanced chapters in mensuration and trigonometry.
Sector vs. Segment vs. Arc
| Concept | What It Contains | Shape |
|---|---|---|
| Sector | Two radii + arc | Pie-slice (like pizza) |
| Segment | Chord + arc | Area between arc and chord |
| Arc | Only the curved edge | Part of circumference |
Try These Yourself
- Find the area of a sector with r = 14 cm and θ = 45°.
- If arc length = 15 cm and r = 12 cm, what is the angle of the sector?
- Calculate the perimeter of a sector where θ = 90° and r = 4 cm.
- Identify whether the shaded region is a major or minor sector (use a diagram from your worksheet).
Frequent Errors and Misunderstandings
- Mixing up sector and segment formulas—remember, a segment uses a chord, a sector uses radii.
- Forgetting that θ must match the unit (degrees for formula above, radians for \( \frac{1}{2} r^2 \theta \)).
- Adding only the arc length (not both radii) when finding sector perimeter.
Applications and Real-Life Use
- Pie charts in statistics and data handling
- Speedometers and dials in vehicles
- Design of garden beds and circular fields
- Calculating the area watered by a rotating sprinkler
Classroom Tip
A quick way to remember the area of a sector: "Fraction of circle area matching the angle." Imagine coloring a part of a coin as the angle opens up — the more the angle, the bigger the sector! Vedantu’s teachers often use pizza or clock visuals in class for easy understanding.
We explored sector of a circle—from definition, formula, types, and stepwise examples to tricks and common mistakes. To become fluent, keep practicing and challenge yourself with different radius and angle values. Vedantu's online resources and interactive sessions make these concepts easy and exam-ready!
Related reads for deeper insight: Area of a Circle, Circumference of a Circle, Segment of a Circle, Arc of a Circle, Trigonometry
FAQs on Sector of a Circle: Definition, Formula, Area & Solved Problems
1. What is a sector of a circle in Maths?
A sector of a circle is a region bounded by two radii and the arc connecting their endpoints. Imagine a slice of pizza; that's a sector! The area of the sector depends on the size of the central angle and the radius of the circle. Key terms include radius, arc, and central angle.
2. How do you calculate the area of a sector?
The area of a sector is calculated using the formula: Area = (θ/360°) × πr², where 'θ' is the central angle in degrees and 'r' is the radius of the circle. Alternatively, if the angle is in radians, the formula is Area = (1/2)r²θ. Remember to use the correct units (square units) for area.
3. What is the difference between a major sector and a minor sector?
A circle is divided into two sectors by any chord. The minor sector is the smaller of the two regions, while the major sector is the larger region. The distinction depends on the size of the central angle; a minor sector has a central angle less than 180°, and a major sector has a central angle greater than 180°.
4. What is the formula for the perimeter of a sector?
The perimeter of a sector is the sum of the lengths of the two radii and the arc length. The formula is: Perimeter = 2r + (θ/360°) × 2πr, where 'r' is the radius and 'θ' is the central angle in degrees. The arc length is calculated as Arc Length = (θ/360°) × 2πr.
5. Where do we use sectors of a circle in real life?
Sectors are used in many real-life applications, including:
- Pie charts: Representing proportions visually.
- Speedometers: Indicating speed with a needle that sweeps across a circular scale.
- Circular gauges: Measuring various quantities such as fuel levels or water pressure.
- Clocks: Each hour marking a different sector.
6. What is the relationship between a sector and the area of a circle?
The area of a sector is a fraction of the total area of the circle. The fraction is determined by the ratio of the central angle of the sector to the total angle of the circle (360°). Specifically, Sector Area / Circle Area = θ/360°.
7. How do you find the area of a sector if only the arc length and radius are given?
If the arc length (l) and radius (r) are known, the area of the sector can be calculated using the formula: Area = (l × r) / 2. This formula is derived from the standard area formula, substituting for the arc length.
8. How is the sector area formula derived?
The area of a sector is derived by considering the proportion of the circle's area that the sector represents. The area of the whole circle is πr². Since a sector subtends an angle θ out of 360°, its area is (θ/360°) multiplied by the total circle area. This leads to the formula: Area = (θ/360°) × πr².
9. What if the sector angle is given in radians?
If the central angle is given in radians, the formula for the sector area simplifies to: Area = (1/2)r²θ, where 'θ' is the angle in radians. This is a more concise form of the formula, often preferred in higher-level mathematics.
10. What is the difference between a sector and a segment of a circle?
A sector is defined by two radii and the arc between them. A segment is the area enclosed by a chord and the arc it subtends. A segment is a portion of the circle cut off by a chord. The key difference is the boundary; a sector is bounded by radii, while a segment is bounded by a chord and an arc.
11. How are sectors used in creating pie charts?
Pie charts use sectors to visually represent proportions of a whole. Each sector represents a category or data point, with its size proportional to the value it represents. The central angle of each sector is calculated as (value/total value) × 360°.
12. Can you explain the concept of a sector in more detail?
A sector, in simpler terms, is a portion of a circle shaped like a pie slice. It's formed by two radii that intersect at the center of the circle and the arc that connects their endpoints. The size of the sector is determined by the central angle, which is the angle formed by the two radii at the center. A larger central angle means a larger sector area.
