Calculate the area of any circle sector quickly and accurately. Whether you're working with semicircles, quadrants, or any other sector size, our calculator handles all cases with precision.
Area = r² × α / 2
where α is in radians
What is a Sector of a Circle?
A sector is a portion of a circle bounded by two radii and an arc, similar to a slice of pizza. The sector's size is determined by its central angle (α) and radius (r).
How to Calculate Sector Area
The area of a sector can be found using these formulas:
where:
- A: Area of the sector
- r: Radius of the circle
- θ: Central angle in radians
Or using degrees:
where:
- A: Area of the sector
- r: Radius of the circle
- θ: Central angle in degrees
Example:
For a sector with radius = 5 inches and angle = 60°:
- A = πr²θ/360°
- A = π × 5² × 60/360
- A = π × 25 × 1/6
- A = 13.1 square inches
Special Cases:
Semicircle (θ = 180°)
For r = 4 inches:
- A = ½ × π × 4²
- A = ½ × π × 16
- A = 25.1 square inches
Quadrant (θ = 90°)
For r = 6 inches:
- A = ¼ × π × 6²
- A = ¼ × π × 36
- A = 28.3 square inches
Real-World Applications
Architecture
Essential for designing arched windows, domed roofs, and calculating material requirements for curved structures.
Engineering
Used in designing gears, rotary mechanisms, and calculating sweep areas for radar systems and robotic arms.
Education
Fundamental concept in geometry, helping students understand circular motion and proportional relationships.
Frequently Asked Questions
How do you find the arc length of a sector?
The arc length (s) can be calculated using: s = (θ × π × r) / 180° where θ is the central angle in degrees and r is the radius.
What's the difference between a sector and a segment?
A sector is bounded by two radii and an arc, while a segment is bounded by a chord and an arc.
How do you convert between degrees and radians?
Multiply degrees by π/180° to get radians, or multiply radians by 180°/π to get degrees.