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 simple terms?
A sector of a circle is the portion of a circle enclosed by two radii and the arc connecting them. A good real-world analogy is a slice of pizza or a pie. The size of the sector is determined by the central angle (θ), which is the angle formed between the two radii at the circle's centre.
2. How do you calculate the area of a sector using the central angle?
The area of a sector is a fraction of the total area of the circle. The formula depends on the unit of the central angle 'θ':
If the angle θ is in degrees, the formula is: Area = (θ/360°) × πr².
If the angle θ is in radians, the formula is: Area = (1/2)r²θ.
Here, 'r' represents the radius of the circle.
3. What is the formula for the perimeter of a sector?
The perimeter of a sector is the total length of its boundary. It is calculated by adding the lengths of the two radii and the length of the arc. The formula is: Perimeter = 2r + l, where 'r' is the radius and 'l' is the arc length. The arc length 'l' itself is calculated as l = (θ/360°) × 2πr.
4. What is the difference between a major sector and a minor sector?
When two radii divide a circle, they form two sectors:
A minor sector is the smaller part, with a central angle less than 180°.
A major sector is the larger part, with a central angle greater than 180°.
Essentially, the minor sector is the smaller "slice," and the major sector is the rest of the circle.
5. How can you find the area of a sector if you only know the arc length and radius?
When the central angle is not given, but you know the arc length (l) and the radius (r), you can use a direct formula to find the area. The formula is: Area = (1/2) × l × r. This is a useful shortcut in many geometry problems, as per the CBSE 2025–26 syllabus.
6. What is the crucial difference between a sector and a segment of a circle?
The key difference lies in how they are formed. A sector is a "pie-slice" shape bounded by two radii and an arc. In contrast, a segment is the region bounded by a chord and the corresponding arc. Think of it this way: a sector is connected to the centre of the circle, while a segment is a piece "cut off" by a straight line (the chord).
7. How is the formula for the area of a sector derived?
The derivation is based on a simple proportion. The area of a full circle (360°) is πr². A sector represents a fraction of that circle, defined by its central angle θ. The ratio of the sector's area to the circle's area is the same as the ratio of its angle to the total angle. This gives us the proportion: (Area of Sector) / (Area of Circle) = θ / 360°. By rearranging, we get the formula: Area of Sector = (θ/360°) × πr².
8. Can a semicircle be considered a sector?
Yes, a semicircle is a special type of sector. A sector is defined by two radii and an arc. In a semicircle, the two radii form a straight line (the diameter), making the central angle exactly 180°. Since it fits the definition of being bounded by two radii and an arc, a semicircle is a sector with a central angle of 180°.
9. How does changing the radius or the central angle affect a sector's area?
The area of a sector is directly proportional to both the central angle and the square of the radius.
If you double the central angle (keeping the radius constant), the area of the sector doubles.
If you double the radius (keeping the angle constant), the area of the sector becomes four times larger because the area depends on the radius squared (r²).
10. What are some real-world examples of sectors of a circle?
Sectors appear in many everyday objects and concepts. Some common examples include:
Pie charts used in data representation.
A single slice of pizza or cake.
The area a windshield wiper cleans on a car's windscreen.
The sweep of a radar on a screen.
The coloured area on a speedometer or fuel gauge.
11. How do you find the central angle of a sector if its area and radius are known?
You can rearrange the standard area formula to solve for the central angle (θ). Using the formula Area = (θ/360°) × πr², you can isolate θ:
θ (in degrees) = (Area × 360°) / (πr²).
This allows you to calculate the angle when the area and radius are given.
12. Why are there two different formulas for the area of a sector?
There are two formulas for convenience, depending on the information given.
Area = (θ/360°) × πr² is used when the central angle (θ) and radius are known.
Area = (1/2) × l × r is used when the arc length (l) and radius are known.
The second formula is derived from the first by substituting the expression for arc length, l = (θ/360°) × 2πr, into the main area formula. They are mathematically equivalent, not different principles.
