Area of Sector Formula: How to Calculate Sector Area in Degrees & Radians
The concept of area of sector plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether it’s solving board exam questions, calculating the area of a pizza slice, or working out parts of a circular park, understanding the area of sector can save you time in calculations and boost your accuracy in competitive tests.
What Is Area of Sector?
An area of sector is defined as the amount of space enclosed by a sector of a circle — essentially, a "slice" bounded by two radii and the arc connecting them. You’ll find this concept applied in geometry questions, mensuration problems, and daily life scenarios such as dividing cakes, pies, or circular fields. Minor and major sectors refer to the smaller and larger parts, depending on the angle at the centre. Understanding this helps in visualising how parts of a circle relate to the whole.
Key Formula for Area of Sector
Here’s the standard formula for calculating the area of sector:
| Situation | Formula | Where |
|---|---|---|
| Angle in Degrees | Area = (θ/360) × π × r² | θ = angle in degrees, r = radius |
| Angle in Radians | Area = (θ/2) × r² | θ = angle in radians, r = radius |
Simply use the formula that matches the units of the angle given in the problem. These formulas are essential for fast calculations in school and entrance exams.
Cross-Disciplinary Usage
Area of sector is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE, NEET, NTSE, or Olympiads will often encounter area of sector in problems involving circular motion, probability, and coding logic for circle-related programs.
Step-by-Step Illustration
Let’s go through an example:
1. Identify what’s given: θ = 60°, r = 7 cm
2. Use the degrees formula: Area = (θ/360) × π × r²
3. Substitute values: Area = (60/360) × π × 7²
4. Simplify: (1/6) × π × 49 = (49/6)π ≈ 25.67 cm²
**Final Answer:** The area of the sector is approximately 25.67 cm².
Speed Trick or Vedic Shortcut
Here’s a quick shortcut: If the question gives the sector as "a part out of n equal parts of the circle", simply divide the area of the entire circle by n to get the sector's area. For example, if a pizza is cut into 8 equal slices, just find the area of the circle and divide by 8—no angle calculation needed!
Example Trick: For semicircles (180°), quadrant (90°), or 1/nth part, use:
| Type | Shortcut Formula |
|---|---|
| Semicircle | ½ × π × r² |
| Quadrant | ¼ × π × r² |
| n Equal Parts | (1/n) × π × r² |
Tricks like these are taught in Vedantu live classes and can boost your calculation speed in MCQs!
Try These Yourself
- Find the area of a sector with r = 10 cm and θ = 45°.
- If a circle of radius 6 cm has a sector area of 18π cm², what is the angle θ in radians?
- Which formula would you use if the angle is given as 2π/3 radians?
- A chocolate pie is cut into 12 equal pieces. What is the area of one piece if the radius is 8 cm?
Frequent Errors and Misunderstandings
- Mixing up degrees and radians in the formula—always check the angle units!
- Forgetting to square the radius (r²) in area calculations.
- Not converting radians to degrees or vice versa when using the wrong formula.
- Missing the central angle or incorrectly dividing the circle.
Relation to Other Concepts
The idea of area of sector connects closely with these topics:
Area of a Circle (basis for sector calculation),
Arc Length (measuring the curved part of the sector),
Area of Segment (when the region is bounded by a chord, not two radii),
and
Area of Parallelogram (compare for mensuration problems).
Linking these will help you easily solve composite figure problems in exams.
Classroom Tip
A quick way to remember area of sector is: “It’s always a fraction of the circle’s area!” Most mistakes are avoided if you write the full formula and substitute carefully. Vedantu’s teachers often draw a pie or pizza slice during class to help you visualize the concept with diagrams and fun examples.
We explored area of sector—from definition, formula, solved examples, frequent mistakes, and connections to other maths topics. Continue practicing with Vedantu and refer to sector worksheets for extra confidence in exams!
FAQs on Area of Sector – Formula, Derivation & Examples
1. What is the area of a sector formula?
The area of a sector is a portion of a circle's area. The formula depends on whether the angle is measured in degrees or radians: * **Degrees:** Area = (θ/360) × πr², where θ is the central angle in degrees and r is the radius. * **Radians:** Area = (1/2)r²θ, where θ is the central angle in radians and r is the radius.
2. How do I calculate the area of a sector when given the arc length and radius?
First, find the central angle (θ). If θ is in radians, use the formula: Arc Length = rθ. Solve for θ. If θ is in degrees use the formula: Arc Length = (θ/360) × 2πr and solve for θ. Then, substitute θ and the radius (r) into the appropriate area formula above.
3. What is the difference between a major and minor sector?
A **minor sector** is the smaller area between two radii and an arc, while a **major sector** is the larger area. The minor sector's central angle is less than 180°, and the major sector's central angle is greater than 180°.
4. How is the area of a sector related to 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 central angle of the sector. For example, a sector with a central angle of 90° represents 1/4 of the circle's area.
5. What are some real-world applications of calculating sector area?
Calculating sector area is useful in many real-world situations, including: * Determining the area of a slice of pizza or pie. * Calculating the area of land represented on a map using a sector of a circle. * Designing parts of circular objects like gears or clock faces. * Analyzing data represented in pie charts.
6. What is a semicircle, and how do I find its area?
A **semicircle** is a sector with a central angle of 180°. Its area is half the area of the circle, calculated as (1/2)πr², where r is the radius.
7. What is a quadrant, and how do I find its area?
A **quadrant** is a sector with a central angle of 90°. Its area is one-quarter of the circle's area, calculated as (1/4)πr², where r is the radius.
8. How do I solve area of sector problems involving shaded regions?
Problems involving shaded regions often require subtracting the area of one sector from another or combining sector and other shapes areas. Break the problem into smaller, manageable areas, calculate each area individually, and then add or subtract to find the final shaded area.
9. What are common mistakes to avoid when calculating the area of a sector?
Common mistakes include: * Using the wrong formula (degrees vs. radians) * Incorrectly converting between degrees and radians * Forgetting to square the radius in the formula * Using the diameter instead of the radius * Making calculation errors
10. What units should I use when calculating sector area?
The units for the radius (r) should be squared; the final area will be expressed in square units (e.g., square centimeters, square meters). The angle (θ) is unitless in the radian formula, and degrees are used in the degree-based formula
11. Can I use a calculator to help with sector area calculations?
Yes, calculators can significantly simplify sector area calculations, especially when dealing with complex angles or large numbers. Ensure the calculator is set to the correct angle mode (degrees or radians), and remember to use parentheses to ensure correct order of operations.
