Sector Area & Arc Length Calculator
Find the area of a circular sector, its arc length, chord and perimeter from the radius and central angle in degrees or radians.
How it works
Angle θ in radians · Sector area = ½ · r² · θ · Arc length = r · θ · Chord = 2r · sin(θ ÷ 2) · Perimeter = 2r + arc
A sector is a pie-slice of a circle bounded by two radii and the arc between them. Working in radians makes everything proportional to the angle: the sector is the fraction θ ÷ 2π of the circle, so its area is that fraction of πr², which simplifies to ½r²θ, and the arc is the same fraction of the circumference 2πr, giving rθ. The straight chord closing the slice depends on the sine of half the angle.
Worked example
For a radius of 6 and a 60° angle, first convert: θ = 60 × π ÷ 180 = π ÷ 3 ≈ 1.0472 radians. The sector area is ½ × 6² × 1.0472 ≈ 18.8496, the arc length is 6 × 1.0472 ≈ 6.2832, and the chord is 2 × 6 × sin(30°) = 12 × 0.5 = 6. The slice is 60 ÷ 360 = one sixth of the circle.
Sources: Circular sector — Wikipedia
Frequently asked questions
Do I have to use radians?
The clean formulas ½r²θ and rθ require radians. This calculator accepts degrees and converts for you with θ = degrees × π ÷ 180, so you can enter whichever you have.
What is the difference between the arc and the chord?
The arc is the curved edge of the slice, of length rθ; the chord is the straight line joining the two ends of that arc, of length 2r·sin(θ ÷ 2). The chord is always shorter than the arc.
What is the sector perimeter?
It is the total distance around the slice: the two radii plus the arc, 2r + rθ. It is not the chord-based boundary — that would describe a circular segment instead.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.