Area of Sector Using Radius and Central Angle
Area of Sector is the region enclosed by two radii and the corresponding arc of a circle. Its area depends on the radius and the central angle. This page covers the area of sector formula, minor and major sectors, special cases such as semicircles and quadrants, solved examples, and useful calculation methods.
What Is the Area of a Sector?
A sector is bounded by two radii and an arc. If θ is less than 180°, it is usually called a minor sector; if θ is greater than 180°, it is a major sector. The angle must be measured at the centre of the circle, and the answer is expressed in square units. For r = 7 cm and θ = 90°, area = (90/360) × π × 7² = 49π/4 cm².
Area of Sector Video Lessons
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Circle Sector and Segment Area
Learn how to calculate the areas of sectors and segments of a circle using their standard formulas and the given radius, angle, or related measurements.
Area of Sector Formula & Tricks
Important Formulas
Use this formula when θ is the central angle in degrees and r is the radius.
Use this formula when θ is the central angle measured in radians.
Subtract the minor sector area from the area of the complete circle.
Use this rearranged formula when the sector area and radius are known.
Here L is the length of the corresponding arc in the same unit as r.
Quick Tricks
The sector area is the same fraction of the full circle area as its central angle is of 360°. First calculate θ/360, then multiply by πr².
For a large sector, calculate the smaller missing sector and subtract it from the full circle. This often reduces the arithmetic.
Area of Sector Concepts
Minor and Major Sectors
For a minor sector, use A = (θ/360°)πr² directly. For a major sector with angle θ, use the same formula or calculate πr² − [(360° − θ)/360°]πr². A sector with angle 180° is a semicircle.
Special Sectors: Quadrant and Semicircle
The area of a quadrant is ¼πr². The area of a semicircle is ½πr². These results follow directly from the general sector formula.
Sector Angle in Radians
The conversion between degrees and radians is θ radians = θ° × π/180. Thus, 60° = π/3 radians. The radian formula avoids dividing by 360 because the angle already represents the arc-to-radius ratio.
Finding an Unknown Radius or Angle
From A = (θ/360°)πr², the radius is r = √(360A/(θπ)), and the angle is θ = 360A/(πr²). Keep the area in square units and the radius in the corresponding linear unit.
Area of Sector Visual Guide & Data
Use the table and chart below to understand the data better.
Common Sector Angles
Area fractions for common central angles of a circle with radius r.
| Central angle | Fraction of full circle | Sector area |
|---|---|---|
| 60° | 1/6 | πr²/6 |
| 90° | 1/4 | πr²/4 |
| 180° | 1/2 | πr²/2 |
| 270° | 3/4 | 3πr²/4 |
Note: These expressions apply to sectors whose radius is r.
A 90° Circular Sector
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Area of Sector: Revision Points
Use the central angle and radius to calculate the required part of a circle.
- For θ in degrees, A = (θ/360°)πr².
- For θ in radians, A = ½r²θ.
- A minor sector has θ < 180°; a major sector has θ > 180°.
- Quadrant area = ¼πr² and semicircle area = ½πr².
- For a major sector, subtract the minor sector area from πr² when that is simpler.
- Area is written in square units, such as cm² or m².
- Use the radius, not the diameter, in the sector formula. If diameter d is given, substitute r = d/2.
Area of Sector FAQs
What is the area of sector formula?
For a central angle θ in degrees, the formula is A = (θ/360°) × πr², where r is the radius.
What is the area of a 90° sector with radius 14 cm?
Area = 90/360 × π × 14² = 49π cm² = 154 cm² when π = 22/7.
How do you find the area of a major sector?
Use A_major = πr² − A_minor, or directly use A_major = (θ/360°)πr² for the major angle.
What is the area of a sector with angle π/3 radians and radius 6 cm?
Using A = ½r²θ, area = ½ × 6² × π/3 = 6π cm².
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc. A segment is bounded by a chord and its corresponding arc, so their area formulas are different.
How is the area of a sector calculated when the diameter is given?
First convert diameter d to radius using r = d/2. Then use A = (θ/360°)π(d/2)².