What Is a Circle Sector?
A circular sector is the region of a circle enclosed by two radii and the arc between them. Its area depends on the circle's radius and the central angle. A larger central angle covers a larger fraction of the circle, while increasing the radius has a squared effect on area.
Use the calculator above with either radius or diameter, and enter the central angle in degrees or radians. The result includes sector area, arc length, sector perimeter, and calculation steps.
Sector Area Formula
When the angle is in degrees:
A = (θ ÷ 360) × π × r²
When the angle is in radians:
A = ½ × r² × θ
Using radius and arc length:
A = ½ × r × s
Here, A is sector area, r is radius, θ is the central angle, and s is arc length.
How to Calculate Sector Area
- Find the radius. If you know the diameter, divide it by 2.
- Identify the central angle. Check whether it is in degrees or radians.
- Choose the correct formula. Use the degree or radian version shown above.
- Substitute the values and calculate.
- Write the answer in square units, such as cm², m², or ft².
A valid sector between 0° and 360° will always have an area greater than 0 and no greater than the full circle area πr².
Degrees vs. Radians
Degrees divide a full revolution into 360 parts, while radians measure angles using the circle's radius. A full revolution equals 360° or 2π radians. Because the formulas use different angle systems, selecting the correct unit is essential.
| Angle Type | Sector Area Formula | Full Revolution |
|---|---|---|
| Degrees | A = (θ/360) × πr² | 360° |
| Radians | A = ½r²θ | 2π radians |
Worked Sector Area Examples
Example 1: 90° Sector
Radius = 5 cm and central angle = 90°.
A = (90/360) × π × 5²
A = 0.25 × π × 25
A = 6.25Ï€
A ≈ 19.635 cm²
A 90° sector is a quadrant, so its area is one quarter of the full circle area.
Example 2: Radian Sector
Radius = 8 m and central angle = 1.5 radians.
A = ½r²θ
A = ½ × 8² × 1.5
A = ½ × 64 × 1.5
A = 48 m²
Common Sector Angles
| Sector | Angle | Fraction of Circle | Area |
|---|---|---|---|
| 30° Sector | 30° | 1/12 | πr² / 12 |
| 60° Sector | 60° | 1/6 | πr² / 6 |
| Quadrant | 90° | 1/4 | πr² / 4 |
| 120° Sector | 120° | 1/3 | πr² / 3 |
| Semicircle | 180° | 1/2 | πr² / 2 |
Major and Minor Sectors
A minor sector has a central angle less than 180°. A major sector has a central angle greater than 180° but less than 360°. A 180° sector is a semicircle, a 90° sector is a quadrant, and a 360° sector is the full circle.
Sector Area, Arc Length, and Perimeter
Sector area measures the two-dimensional space inside the sector. Arc length measures the curved part of the boundary, while sector perimeter is the arc length plus the two radii.
Arc length in radians: s = rθ
Arc length in degrees: s = (θ/360) × 2πr
Sector perimeter: P = s + 2r
For broader boundary calculations, use the Perimeter Calculator. For general surface measurements, use the Area Calculator.
Common Circle Sector Mistakes
- Using the wrong angle formula: Degrees and radians use different formulas.
- Confusing diameter and radius: Radius is half the diameter.
- Forgetting to square the radius: Sector area depends on r².
- Using linear units for area: Sector area must be written in square units.
- Entering more than one full revolution: Standard sector inputs should not exceed 360° or 2π radians.
How to Check Your Answer
First calculate the full circle area using πr². A sector with an angle between 0° and 360° should have an area between 0 and the full circle area. A 90° sector should equal one quarter of the circle area, while a 180° sector should equal one half.
You can also check whether the arc length and sector perimeter are reasonable compared with the full circumference 2Ï€r.
Real-World Example: Pizza Slice
A 14-inch pizza has a radius of 7 inches and is cut into 8 equal slices. Each slice has a central angle of 45°.
A = (45/360) × π × 7²
A = 1/8 × 49π
A ≈ 19.242 in²
Each slice covers one eighth of the total pizza area.
Frequently Asked Questions
For degrees, use A = (θ/360) × πr². For radians, use A = ½r²θ.
Yes. Divide the diameter by 2 to get the radius, then use the appropriate sector area formula. The calculator above does this conversion automatically when Diameter is selected.
A sector is bounded by two radii and an arc. A circular segment is bounded by a chord and an arc.
A minor sector has a central angle less than 180°.
A major sector has a central angle greater than 180° but less than 360°.
If radius r and arc length s are known, use A = ½rs.
A semicircle has a 180° central angle, so its area is πr²/2.
A quadrant is a 90° sector, so its area is πr²/4.
Calculation methodology
The calculator uses the standard circular-sector formulas for degrees and radians. If diameter is selected, radius is calculated as diameter ÷ 2. JavaScript's built-in numerical approximation of π is used, and rounding is applied only to the displayed result.
About this calculator: Maintained by Utilixea. Formulas, validation rules, and example calculations are checked whenever the tool is updated.
Need another geometry calculation? Explore our Geometry Calculators or try the Area Calculator, Perimeter Calculator, and Distance Between Points Calculator.
