Area of a Sector Calculator
Calculate the area, arc length, and chord length of a circular sector from its radius and central angle. See also Arc Length Calculator and Area of Circle Calculator.
How to Calculate the Area of a Sector
A sector is a "pie slice" of a circle defined by two radii and the arc between them. To find the area, you need the radius and the central angle. If the angle is in degrees, use A = (θ/360) × πr². If in radians, use A = ½r²θ. The arc length is the curved edge of the sector, and the chord length is the straight line connecting the two endpoints of the arc.
Sector Area Formula
A = ½ × r² × θ (θ in radians)
A = (θ/360) × π × r² (θ in degrees)
Arc Length = r × θ (radians)
Chord Length = 2r × sin(θ/2)
Example
Find the area of a sector with radius 10 and angle 60°:
A = (60/360) × π × 10²
A = (1/6) × π × 100
A ≈ 52.3599 square units
Arc Length = 10 × (π/3) ≈ 10.4720
Chord Length = 2 × 10 × sin(30°) = 10
Sector Area Reference Table
| Radius | Angle (°) | Area | Arc Length |
|---|---|---|---|
| 5 | 30° | 6.5450 | 2.6180 |
| 5 | 60° | 13.0900 | 5.2360 |
| 5 | 90° | 19.6350 | 7.8540 |
| 10 | 30° | 26.1799 | 5.2360 |
| 10 | 60° | 52.3599 | 10.4720 |
| 10 | 90° | 78.5398 | 15.7080 |
| 10 | 120° | 104.7198 | 20.9440 |
| 10 | 180° | 157.0796 | 31.4159 |
| 10 | 270° | 235.6194 | 47.1239 |
| 10 | 360° | 314.1593 | 62.8319 |
| 15 | 45° | 88.3573 | 11.7810 |
| 20 | 60° | 209.4395 | 20.9440 |
When to Use the Sector Area Formula
Use the sector area formula whenever you need to find the area of a "pie slice" shape — a wedge cut from a circle. Common situations include:
- Pie and pizza slices — determining the serving area of each slice based on how many pieces you cut
- Windshield wiper coverage — the swept area of a wiper blade forms a sector (or annular sector)
- Camera field of view — security cameras with a known angle of coverage scan a sector-shaped region
- Circular garden sections — dividing a circular garden into wedge-shaped planting zones
- Fan and propeller blades — each blade sweeps through a sector as it rotates
- Radar coverage — military and weather radar scan sector-shaped areas
The sector is defined by two radii and the arc between them. The key inputs are the radius and the central angle (in degrees or radians).
Solved Examples
Example 1: Pizza Slice Area
A 14-inch (diameter) pizza is cut into 8 equal slices. What is the area of each slice?
Radius = 14/2 = 7 inches
Angle per slice = 360°/8 = 45°
A = (45/360) × π × 7²
A = (1/8) × π × 49
A ≈ 19.24 square inches per slice
Example 2: Windshield Wiper Swept Area
A windshield wiper is 45 cm long and sweeps through an angle of 120°. What area does it clean?
Radius = 45 cm, Angle = 120°
A = (120/360) × π × 45²
A = (1/3) × π × 2025
A = (1/3) × 6,361.73
A ≈ 2,120.58 cm² of glass cleaned
Example 3: Security Camera Coverage
A security camera has a 90° field of view and can see up to 20 meters away. What ground area does it cover?
Radius = 20 m, Angle = 90°
A = (90/360) × π × 20²
A = (1/4) × π × 400
A = (1/4) × 1,256.64
A ≈ 314.16 square meters of coverage
Common Mistakes When Calculating Sector Area
Mixing degrees and radians
The formula A = ½r²θ requires θ in radians. If your angle is in degrees, use A = (θ/360) × πr² instead. Using 90 (degrees) directly in the radian formula gives a hugely wrong answer.
Using diameter instead of radius
A 14-inch pizza has a 7-inch radius, not 14. Using the diameter in the formula gives 4× the correct sector area.
Confusing sector area with arc length
Sector area (A = ½r²θ) gives square units. Arc length (L = rθ) gives linear units. They answer different questions: how much surface vs. how far along the curve.
Key Takeaways
- Sector area in radians: A = ½r²θ. In degrees: A = (θ/360) × πr².
- A sector is a fraction of the full circle: the fraction equals θ/360° (or θ/2π in radians).
- Arc length of the sector: L = rθ (radians) or L = (θ/360) × 2πr (degrees).
- A semicircle is a sector with θ = 180°, giving area = πr²/2.
- A quarter circle (quadrant) has θ = 90°, giving area = πr²/4.
- To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π.
Frequently Asked Questions
What is a sector?
A sector is a region of a circle enclosed by two radii and the arc between them — like a slice of pizza or pie. The central angle determines what fraction of the full circle the sector represents.
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc (pie slice shape). A segment is bounded by a chord and an arc. The segment area = sector area − triangle area formed by the two radii and the chord.
How do I convert between degrees and radians?
Multiply degrees by π/180 to get radians. Multiply radians by 180/π to get degrees. For example, 90° = π/2 radians, and π radians = 180°.
What is the area of a semicircle?
A semicircle is a sector with angle 180° (π radians). Its area is half the full circle: A = πr²/2.