Arc Length and Sector Area Formulas
The arc length formula is L = (θ ⁄ 360) × 2πr, and the area of a sector is A = (θ ⁄ 360) × πr². Both use one idea: take the fraction of the full circle that the angle covers.
Sector of a Circle
A sector is a “pizza slice” of a circle bounded by two radii and an arc. Its share of the circle equals θ ⁄ 360°.
- Area
- A = (θ ⁄ 360°) × πr2
- Arc length
- L = (θ ⁄ 360°) × 2πr
- r radius
- θ central angle in degrees
The fraction idea
A full turn is 360°. A 90° sector is 90 ⁄ 360 = ¼ of the circle, so its arc is ¼ of the circumference and its area is ¼ of the circle’s area.
Worked example
A sector has radius 6 cm and angle 60°.
- Fraction = 60 ÷ 360 = 1⁄6
- Arc length = 1⁄6 × 2π × 6 = 2π = 6.28 cm
- Sector area = 1⁄6 × π × 6² = 6π = 18.85 cm²
- Perimeter of the sector = 6.28 + 6 + 6 = 18.28 cm
In radians
If the angle θ is in radians, the formulas are simpler: L = rθ and A = ½r²θ.
Try the sector calculator
Try it yourself
A pizza slice has radius 15 cm and angle 45°. How long is its crust?
Show the answer
L = (45 ⁄ 360) × 2π × 15 = 11.78 cm
More on the sector of a circle formulas page.
Arc length and sector area: FAQ
What is the arc length formula?
L = (θ ⁄ 360) × 2πr in degrees, or L = rθ in radians.
What is the area of a sector?
A = (θ ⁄ 360) × πr² in degrees, or A = ½r²θ in radians.
What is the perimeter of a sector?
The arc length plus two radii: P = L + 2r.