What is the area of a sector?
A sector is a slice of a circle made by two radii and the arc between them. Think of a pizza slice: if you know the radius and the central angle, you can calculate exactly how much surface area that slice covers.
The area of a sector is proportional to the angle. If the angle is half of a full circle, the sector takes half the circle’s area. If the angle is one quarter of a full circle, the sector area is one quarter of the full area, and so on.
Sector area formulas
When angle is in degrees
Area = (θ / 360) × π × r²
- θ = central angle in degrees
- r = radius of the circle
When angle is in radians
Area = 1/2 × r² × θ
- θ = central angle in radians
- r = radius of the circle
How to use this area of a sector calculator
- Enter the circle’s radius.
- Enter the central angle value.
- Choose whether the angle is in degrees or radians.
- Click Calculate to get the sector area instantly.
The tool also shows the fraction of the full circle and the arc length, which is useful in geometry, engineering, and design problems.
Worked examples
Example 1: Degrees
Suppose r = 10 and θ = 90°. A full circle area is π × 10² = 100π. Since 90° is one quarter of 360°, the sector area is:
(90/360) × 100π = 25π ≈ 78.54 square units.
Example 2: Radians
Suppose r = 6 and θ = 1.2 rad. Using the radians formula:
Area = 1/2 × 6² × 1.2 = 21.6 square units.
Common mistakes to avoid
- Using degrees in the radians formula (or vice versa).
- Forgetting to square the radius.
- Using diameter instead of radius by accident.
- Ignoring units (cm², m², in², etc.) in final answers.
Why this calculator is useful
Sector calculations appear in many real-world situations: architecture curves, sprinkler coverage, pie-chart geometry, wheel mechanics, and circular land plots. This calculator helps you move quickly from raw inputs to a clear result without manual conversion errors.
FAQ
Can the angle be larger than 360°?
Yes. The calculator still computes the mathematical area based on your input. An angle greater than 360° represents more than one full revolution.
Can I use decimals?
Absolutely. Decimal radius and angle values are fully supported.
What units are used?
The calculator is unit-agnostic. If your radius is in meters, the result is in square meters. If your radius is in inches, the area is in square inches.